Theta-vexillary signed permutations

Theta-vexillary signed permutations are elements in the hyperoctahedral group that index certain classes of degeneracy loci of type B and C. These permutations are described using triples of $s$-tuples of integers subject to specific conditions. The objective of this work is to present different characterizations of theta-vexillary signed permutations, describing them in terms of corners in the Rothe diagram and pattern avoidance.


Introduction
A permutation w is called vexillary if and only if it avoids the patters r2 1 4 3s, i.e., there are no indices a ă b ă c ă d such that wpbq ă wpaq ă wpdq ă wpcq. Vexillary permutations were found by Lascoux and Schützenberger [10] in the 1980s. Fulton [8] in the 1990s obtained other equivalent characterizations for the vexillary permutations: in addition to the pattern avoidance criterion, he figured out one in terms of the essential set of a permutation, among others. Since S n is the Weyl group of type A, the vexillary permutations represent Schubert varieties in some flag manifold where the Lie group is G " Slpn, Cq.
A few years later, the notion of vexillary permutations in the hyperoctaedral group were introduced by Billy and Lam [5]. Recently, Anderson and Fulton [1,4] provided a different characterization for vexillary signed permutations. They defined them through a specific triple of integers: given three s-tuple of positive integers τ " pk, p, qq, where k " p0 ă k 1 ă¨¨¨ă k s q, p " pp 1 ě¨¨¨ě p s ą 0q, and q " pq 1 ě¨¨¨ě q s ą 0q, satisfying p i´pi`1`qi´qi`1 ą k i`1´ki for 1 ď i ď s´1, one constructs a signed permutation w " wpτ q. Since the hyperoctahedral group W n can be included in the group S 2n`1 , a signed permutation w in W n is vexillary if and only if its inclusion ιpwq in S 2n`1 is a vexillary permutation as mentioned above. Anderson and Fulton in [4] characterize vexillary signed permutations in terms of essential sets, pattern avoidance, and Stanley symmetric functions.
In this work, we present a class of signed permutations called theta-vexillary signed permutations. They are defined using a triple of integers τ " pk, p, qq where we allow negative values for q and satisfy eight different conditions, which will be called a theta-triple. The set of theta-vexillary signed permutations is relevant because it contains all vexillary signed permutations and k-Grassmannian permutations, which are the ones associated to the Grassmannian Schubert varieties of type B and C.
Theta-vexillary signed permutations have an important geometric interpretation in terms of degeneracy loci. For our purpose, it is easier to denote the hyperoctahedral group as the Weyl group of type B. Consider a vector bundle V of rank 2n`1 over X, equipped with a nondegenerate form and two flags of bundles E ‚ " pE p1 Ă E p2 Ă¨¨¨Ă E ps Ă V q and F ‚ " pF q1 Ă F q2 Ă¨¨¨Ă F qs Ă V q such that: for q ą 0, the subbundles F q are isotropic, of rank n`1´q; for q ă 0, F q is coisotropic, of corank n`q; and all the subbundles E p are isotropic, of rank n`1´p. The degeneracy locus of τ is Ω τ :" tx P X | dimpE pi X F qi q ě k i , for 1 ď i ď su.
Anderson and Fulton in [2] figured out that if the triple τ is subjet to certain conditions, the cohomology class rΩ τ s is the multi-theta-polynomial Θ λpτ q whose coefficiens are Chern classes of the vector bundles E pi and F qi . The polinomial Θ λpτ q derives from the theta-polynomials defined via raising operators by Bush, Kresch, and Tamvakis [7] and inspired the name theta-vexillary signed permutations.
The main result of this work provides two other ways to characterize thetavexillary permutations. If a permutation w in the Weyl group W n of type B is represented as a matrix of dots in a p2n`1qˆn array of boxes, the (Rothe) extended diagram is the subset of boxes that remains after striking out the boxes weakly south or east of each dot. The southeast (SE) corners in the extended diagram form the set of corners C pwq. One characterization of theta-vexillary signed permutations is the set of corners C pwq is the disjoint union of the set Nepwq which is composed by all corners that form a piecewise path that goes to the northeast direction, and the set Upwq of unessential corners. We also have a characterization via pattern avoidance.
This theorem is consequence of Propositions 15 and 18 and it is similar to the vexillary signed permutation's version. It is interesting to notice that, comparing to the vexillary case, we admit some SE corners in the diagram that are not in an ordered northeast path, which we call the unessential corners. Besides, the characterization via signed pattern avoidance for the theta-vexillary permutations has eight patterns in common with those for the vexillary case and r2 1s is the unique not present in this list.
Considering the pattern avoidance criterion, the set of theta-vexillary signed permutations form a new class of permutations according to the "Database of Permutation Pattern Avoidance" maintained by Tenner [11].
This work is part of my Ph.D. thesis [9]. this work while I was a visiting scholar at The Ohio State University. I also thank to Lonardo Rabelo for comments on a previous manuscript.

Signed permutations in W n
The notation present here is the same used in [3]. We also refer [6, §8.1] for further details.
A signed permutation is a permutation w satisfying that wpıq " wpiq, for each i. A signed permutation belongs to W n if wpmq " m for all m ą n; this is a group isomorphic to the hyperoctahedral group, the Weyl group of types B n and C n . Since wpıq " wpiq, we just need the positive positions when writing signed permutation in one-line notation, i.e., a permutation w P W n is represented by wp1q wp2q¨¨¨wpnq. For example, the full form of the signed permutation w " 2 1 3 in W 3 is 3 1 2 0 2 1 3, but we can omit the values at the position 3, 2, 1 and 0. The group W n is generated by the simple transpositions s 0 , . . . , s n , where for i ą 0, right-multiplication by s i exchanges entries in positions i and i`1, and right-multiplication by s 0 replaces wp1q with wp1q. Every signed permutation w can be written as w " s i1¨¨¨si ℓ such that ℓ is minimal; call the number ℓ " ℓpwq the length of w. This value counts the number of inversions of w P W n , and it is given by the formula ℓpwq " #t1 ď i ă j ď n | wpiq ą wpjqu`#t1 ď i ď j ď n | wp´iq ą wpjqu. (2.1) The element w pnq " 1 2¨¨¨n is the longest element in W n and it is called the involution of W n . Notice that the involution w pnq has length n 2 . The group of permutations W n can be embedded in the symmetric group S 2n`1 , considering S 2n`1 the permutations of n, . . . , 0, . . . , n. Indeed, define the odd embedding by ι : W n ãÑ S 2n`1 where it sends w " wp1q wp2q¨¨¨wpnq to the permutation wpnq¨¨¨wp2q wp1q 0 wp1q wp2q¨¨¨wpnq in S 2n`1 . The embedding ι will be used when it is necessary to highlight that we need the full permutation of w.
There is also a even embedding ι 1 : W n ãÑ S 2n defined by omitting the value wp0q " 0.
Considering the natural inclusions W n Ă W n`1 Ă¨¨¨, we get the infinite Weyl group W 8 " YW n . When the value n is understood or irrelevant, we can consider w as an element of W 8 . The odd embeddings are compatible with the corresponding inclusions S 2n`1 Ă S 2n`3 Ă¨¨¨.

2.1.
Diagram of a permutation in S 2n`1 . Let us consider the specific case where the permutation group is S 2n`1 . It is important to consider this case because we need to do some modification in the notation that will be useful for us.
Consider a p2n`1qˆp2n`1q arrays of boxes with rows and columns indexed by integers rn, ns " tn, . . . , 1, 0, 1, . . . , nu in matrix style. The permutation matrix associated to a permutation w P S 2n`1 is obtained by placing dots in positions pwpiq, iq, for all n ď i ď n, in the array. Again the diagram of w is the collection of boxes that remain after removing those which are (weakly) south and east of a dot in the permutation matrix. Observe that the number of boxes in the diagram is equal to the length of the permutation.
The rank function of a permutation w P S 2n`1 for a pair pp, qq, where n ď p, q ď n, is the number of dots strictly south and weakly west of the box pq´1, pq in the permutation matrix of w. In other words, it will be defined by r w pp, qq :" #ti ď p | wpiq ě qu, for n ď p, q ď n.
We say that a box pa, bq is a southeast (SE) corner of the diagram of w if w has a descent at b, with a lying in the interval of the jump, and w´1 has a descent at a, with b lying in the interval of the jump. This can be written as wpbq ą a ě wpb`1q and A corner position of w is a pair pp, qq such that the box pq´1, pq is a southeast (SE) corner of the diagram of w. The set of corners of w is the set C pwq of triples pk, p, qq such that pp, qq is a corner position and k " r w pp, qq.
For example, consider w " ιp2 3 1q " 1 3 2 0 2 3 1. Figure 1 shows the diagram of w. The SE corners pq´1, pq are highlighted and they are filled with the rank function values r w pp, qq. In this case, the set of corners is C pwq " tp1, 3, 1q, p1, 1, 2q, p3, 0, 1q, p2, 2, 2qu. Notice that if a box pq´1, pq is a SE corner that satisfies (2.2), then pp, qq is a corner position and k " r w pp, qq.

2.2.
Extended diagram of a signed permutation in W n . We know that signed permutations must satisfy the relation wpıq " wpiq, then the negative positions can be obtained from the positive ones. Hence, a signed permutation w P W n corresponds to a p2n`1qˆn array of boxes, with rows indexed by tn, . . . , nu and the columns indexed by tn, . . . , 1u, where the dots are placed in the boxes pwpiq, iq for n ď i ď 1.
For each dot, we place an "ˆ" in those boxes pa, bq such that a " wpiq and i ď b, in other words, anˆis placed in the same column and opposite along with the boxes to the right of thisˆ.
The extended diagram D`pwq of a signed permutation w is the collection of boxes in the p2n`1qˆn rectangle that remain after removing those which are south or east of a dot. The diagram Dpwq Ď D`pwq is obtained from extended diagram D`pwq by removing the ones marked withˆ. Namely, Dpwq is defined by Dpwq " tpi, jq P rn, 1sˆrn, ns | wpiq ą j, w´1pjq ą i, and w´1p´jq ą iu. Proof. Observe that if we define J " tj P rn, ns | w´1pjq ă 0u, the set Dpwq can be split into two subsets D 1 pwq " tpi, jq P Dpwq | j P Ju and D 2 pwq " tpi, jq P Dpwq | j R Ju. Since both sets have cardinality #D 1 pwq " #t1 ď l ă m ď n | wplq ą wpmqu and #D 2 pwq " #t1 ď l ď m ď n | wp´lq ą wpmqu, the assertion follows from Equation (2.1).
Observe that if we use the embedding ι : W n ãÑ S 2n`1 , the matrix and extended diagram of w P W n corresponds, respectively, to the first n columns of the matrix and diagram of ιpwq. The notation ιpD`pwqq will be used when we need to use the respective p2n`1qˆp2n`1q diagram of ιpwq.
The rank function of a permutation w in W n is defined by for 1 ď p ď n, and n ď q ď n.
Since wpıq " wpiq, then the rank function r w pp, qq is also equal to #ti ď p | wpiq ě qu, so the rank functions r w coincides to r ιpwq .
Given w P W n , the next lemma states that there is a symmetry about the origin of the corner positions corresponding to ιpwq. In order to simplify the notation, given a triple pk, p, qq, define the reflected triple pk, p, qq K " pk`p`q´1, p`1, q`1q.

Lemma 3 ([3]
, Lemma 1.1). For w P W n , the set of corners of ιpwq P S 2n`1 has the following symmetry: pk, p, qq is in C pιpwqq if and only if pk, p, qq K is in C pιpwqq.
We can see in Figure 1 that both corners in the left half of the diagram are symmetric by the origin to other two corners in the right side. This behavior will happen for every signed permutation w, implying that half of C pιpwqq suffices to determine the signed permutation w; we will consider those corners appearing in the first n columns.
A corner position of signed permutation w is a pair pp, qq such that the box pq´1, pq is a southeast (SE) corner of the extended diagram of w. The set of corners C pwq of a signed permutation w is the set of triples pk, p, qq such that pq´1, pq is a SE corner of the extended diagram D`pwq and k " r w pp, qq, except for corner positions pp, qq where p " 1 and q ă 0. This exception comes from the fact that p1, qq, for q ă 0, is not a corner position in ιpwq because the respective box pq´1, 1q cannot be a SE corner since wp0q " 0. [3,4] defined a slightly different set called essential set of w. This set is contained in the set of corners, since they remove a few "redundant" SE corners. In the present work, we need the whole set of corners since the essential set is not enough to perform our computations.

Remark 4. Anderson and Fulton in
Since the integer k is the rank of w in pp, qq, sometimes we can simply say that the corner position pp, qq P C pwq, instead of the triple pk, p, qq.
The Figure 2 illustrates the extended diagram and the set of cornet of the signed permutation w " 10 1 5 3 2 4 6 9 8 7. To make the diagrams look cleaner, from now on we won't denoteˆin the extended diagrams D`pwq. During the text, it can happen that we omit the word "extended" since we are only interested in studying the extended diagram of a signed permutation w so that the diagram Dpwq won't be useful for us.
2.3. NE path and unessential corners. Suppose that w P W n is any signed permutation. There are two notable classes of SE corner in the set C pwq that we will be important to our main theorem. They are the corners in the northeast path and the unessential corners.
Given any signed permutation w, consider a (strict) partial order for the set of corners C pwq by pp, qq ă pp 1 , q 1 q if and only if p ą p 1 and q ă q 1 , for corner positions pp, qq, pp 1 , q 1 q P C pwq. For example, in Figure 2, the unique possible relation is p4, 3q ă p2, 2q, the two boxes filled in with the value 6.
Define the northeast (NE) path as the set Nepwq of minimal elements of C pwq relative to the poset "ă". Using the same example of Figure 2, we have that Nepwq " C pwq´tp6, 2, 1qu, since all the corners are minimal under this poset except p6, 2, 1q.
The positions pp i , q i q of the NE path Nepwq can be ordered so that p 1 ě p 2 ě¨¨ě p r ą 0 and q 1 ě q 2 ě¨¨¨ě q r . In fact, suppose that we order p 1 ě p 2 ě¨¨¨ě p r ą 0 but there is i such that and q i ă q i`1 . If p i " p i`1 then we can exchange i and i`1. Otherwise, if p i ą p i`1 then pp i , q i q ă pp i`1 , q i`1 q and pp i`1 , q i`1 q does not belong to the NE path.
Given a signed permutation w, we say that a corner position pp, qq of C pwq is unessential if there are corner positions pp 1 , q 1 q, pp 2 , q 2 q and pp 3 , q 3 q in the NE path Nepwq satisfying the following conditions: p 1 " p and q 1 ă q ă 0; p 2 ą 0 and q 2 " q`1; In other words, pp, qq is not a minimal corner in the poset in the upper half of the diagram, the box pq 1´1 , p 1 q lies above and in the same column of the box pq´1, pq, and the box pq 2 , p 2´1 q reflected from pq 2´1 , p 2 q lies to the right and in the same row of pq´1, pq, as shown in Figure 3. We denote by Upwq the set of all unessential corners of w. It is important to emphasize that all three corners pp 1 , q 1 q, pp 2 , q 2 q and pp 3 , q 3 q must belong to the NE path Nepwq.

Theta-triples and theta-vexillary signed permutations
A theta-triple is three s-tuples τ " pk, p, qq with k " p0 ă k 1 ă k 2 ă¨¨¨ă k s q, p " pp 1 ě p 2 ě¨¨¨ě p s ą 0q, (3.1) q " pq 1 ě q 2 ě¨¨¨ě q s q, and satisfying eight conditions. We intentionally split such conditions in three blocks that share common characteristics. The first three conditions are A1. q i ‰ 0 for all i; A2. q i ‰´q j , for any i ‰ j. A3. If q s ă 0 then p s ą 1; Now, let a " apτ q be the integer such that q a´1 ą 0 ą q a , allowing a " 1 and a " s`1 for the cases where all q's are negative or all q's are positive, respectively. For all i ě a, denote by Rpiq (or Rpiq τ to specify the triple) the unique integer such that q Rpiq ą´q i ą q Rpiq`1 ; if necessary, consider k 0 " 0, p 0 "`8, q 0 "`8, and Rpa´1q " a´1. The next three conditions are It is important to observe that none of the above conditions compare indexes a´1 and a. Finally, consider a ď i ď s and let Lpiq " L τ piq be the biggest integer j in tRpiq`1, . . . , a´1u satisfying k j´kRpiq`1 ě q Rpiq`1´qj , i.e., The last two conditions are C1.´q i ě k i´kRpiq for all a ď i ď s; C2.´q i ě q Lpiq`kLpiq´kRpiq for all a ď i ď s. Given a theta-triple τ , the construction algorithm of the permutation wpτ q is given by a sequence of s`1 steps as follows: Step (1): Starting in the p 1 position, place k 1 consecutive entries, in increasing order, ending with´q 1 . Mark the absolute value of these numbers as "used"; Step (i): For 1 ă i ď s, starting in the p i position, or the next available position to the right, fill the next available k i´ki´1 positions with entries chosen consecutively from the unused absolute numbers, in increasing order, ending with´q i or, if it is not available, the biggest unused number beloẃ q i . Again, mark the absolute value of these numbers as "used"; Step (s`1): Fill the remaining available positions with the unused positive numbers, in increasing order. Notice that we should mark as used the absolute of the placed values because we allow negative q i for a theta-triple.
Notice the construction algorithm does not create an inversion inside a step, i.e., if a ă b are positions placed by a Step piq then wpaq ă wpbq.
Remark 5. The definition of a theta-triple was motivated by the triple of type C given by Anderson and Fulton [2]. Indeed, any theta-triple is a triple of type C, but the converse is not true. A theta-triple has two properties that a triple of type C does not: each pk i , p i , q i q is associated to a SE corner of wpτ q (Proposition 10); and any theta-vexillary permutation is given by a unique theta-triple τ (Proposition 17). Both results are relevant when we study SE corners in the diagram of wpτ q. Now, we will give a brief explanation about the eight conditions of a theta-triple. Conditions A1, A2 and A3 guarantee that the permutation wpτ q associated to such theta-triple is a signed permutation.
Conditions B1, B2 and B3, in some sense, characterize a theta-vexillary permutations as well as the condition pp i´pi`1 q`pq i´qi`1 q ą k i`1´ki does for vexillary permutations. In condition B2, an extra k Rpiq´kRpi`1q is added to the right side because, during the construction of wpτ q, Step piq skips an equal number of entries that have already been used in previous steps. Moreover, condition B3 is equivalent to apply i " s in condition B2, where we consider the extreme cases pk 0 , p 0 , q 0 q " p0, n, nq and pk s`1 , p s`1 , q s`1 q " pn, 1,´nq.
Finally, for conditions C1 and C2, the next lemma states an equivalent definition of them based on the construction algorithm of wpτ q: Lemma 6. The conditions C1 and C2 are equivalent, respectively, to C1 1 . Given any a ď i ď s, all entries placed by Steps paq to piq are positive; C2 1 . Given any a ď i ď s, all entries placed by Steps pRpiq`1q to pa´1q are strictly bigger than q i .
Proof. For the first statement, observe that all Steps from paq to piq must skip at most k a´1´kRpiq values because they were already used in Steps pRpiq`1q to pa´1q and denote by α :"´q i´p k a´1´kRpiq q the number of available positive entries from 1 to q i that can be used by Steps paq to piq. Then, condition C1 is equivalent to say that α ě k i´ka´1 , i.e., there is enough positive values available to be placed by Steps paq to piq.
For the second assertion, remember that the definition of Lpiq says that it is the biggest integer in tRpiq`1, . . . , a´1u where k Lpiq´kRpiq`1 ě q Rpiq`1´qLpiq . The smallest possible entry placed by Steps pRpiq`1q to pLpiqq is limited below by q Lpiq`kLpiq´kRpiq`1 . Since for any Step pjq after Lpiq, we have that k jḱ Rpiq`1 ă q Rpiq`1´qj , then no entry placed by such step cannot be smaller than q Lpiq . So, every entry placed by Steps pRpiq`1q to pa´1q is limited below by q Lpiq`kLpiq´kRpiq`1 , and we conclude that both conditions C2 and C2 1 imply that q i ă q Lpiq`kLpiq´kRpiq`1 .
In other words, conditions C1 and C2 guarantee that given i ě a, then all values placed by Steps pRpiq`1q to piq ranges from q i to q i . In practice, it will be easier to use C1 1 and C2 1 instead of C1 and C2. Now, let us study the descents of a theta-vexillary signed permutation wpτ q and its inverse wpτ q´1. Proposition 7. Let w " wpτ q be a theta-vexillary signed permutation and τ " pk, p, qq be a theta-triple. Then all the descents of w are at positions p i´1 , i.e, for each i, we have wpp i´1 q ą q i ě wpp i q and there are no other descents.
Proof. In Step p1q, no descents are created, unless p 1 " 1, in which case the permutation has a single descent at 0. For 1 ă i ă a, this is proved in Lemma 2.2 of [4]. Now, supposing that a ď i ď s and i ě 2, assume inductively that for j ă i, there is a descent at position p j´1 whenever this positions has been filled, satisfying wpp j´1 q ą q j ě wpp j q, and there are no other descents. By Lemma 6, only positive entries are placed in consecutive vacant positions of Step piq, from left to right, at position p i (or the next vacant position to the right, if p i´1 " p i ). We consider "sub-steps" of Step piq, where we are placing an entry at position p ě p i , and distinguish three cases. First, suppose we are at position p, with p ă p i´1´1 . In this case, the previous entry placed in Step piq (if any) was placed at position p´1, so we did not create a descent at p´1. Position p`1 is still vacant, so no new descents are created.
To clarify this proof, let τ " p3 4 5 6 9, 8 6 5 4 2, 7 4 2 3 6q as in Example 1. In Step p5q, the first entry placed is 1 and it does not create a descent: w "¨1¨3 2 4¨9 8 7 Next, suppose we are at position p " p i´1´1 . This means that p i´1´pi ď k i´ki´1 , so let β " pk i´ki´1 q´pp i´1´pi q be the number of entries remaining to be placed in Step piq, after placing the current at position p. Condition B1 tell us that q i ď q i`β ă q i´1 , then considering the integer interval I i " tq i´1`1 , . . . , q i u, it must be non-empty. We claim that the entry wppq " wpp i´1´1 q lies in I i and therefore wpp i´1´1 q ą q i´1 ě wpp i´1 q, proving this situation. Remember that the construction algorithm must skip those entries that its absolute value have already been used, and then this claim is equivalent to say that even removing from I i those repetitions, there still is some value to be picked by wppq in I i .
To prove that claim, lets count how many values in I i were used in previous steps. For a ď j ă i, any entry x of Steps pjq satisfies x ď q j ď q i´1 , that means x R I i . If 1 ď j ď Rpiq then any entry x placed in Steps pjq satisfies x ď q j ď q Rpiq ă q i , implying that x R I i . If Rpi´1q ă j ă a then by condition C2 1 , any entry x placed in Steps pjq satisfies x ą q i´1 , implying that x R I i . Finally, if Rpiq ă j ď Rpi´1q then by condition C2 1 , any entry x placed in Step pjq satisfies q i ă x ď q j ď q Rpi´1q ă q i´1 , hence, x P I i . We conclude that the only absolute values placed in previous steps that belongs to the interval I i are all the ones from Steps pRpiq`1q to pRpi´1qq. So there are α :" k Rpi´1q´kRpiq values in I i that cannot be used in Step piq in position p. In order to place the correct value for position p of Step piq, we need to consider that the values which are going to be placed after position p also must belong to I i and are bigger than wppq, i.e., it also is required to skip the β biggest values in I i . Since the number of elements of I i is q i´qi´1 , follows from condition B2 that #pI i q ą α`β and, therefore, there is some value in I i to pick for wppq.
Continuing the example, in Step p5q, the second entry placed is 5, which creates a descent at position 3: w "¨1 5 3 2 4¨9 8 7 Finally, suppose we are at position p ě p i´1 . Using the previous case, the entry to be placed is some x P I i . When an entry is placed in a vacant position to the right of a filled position, it does not create a descent since either all entries already placed the previous steps are smaller than q i´1 ă x or the entries placed in this step is smaller than x. When it is placed to the left of a filled position, which can only happen at positions p j´1 for some j ă i´1, and it does create a descent at the position p j´1 satisfying wpp j´1 q ą q i´1 ě q j ě wpp j q In Step p5q of our example, it remains to place the 3rd value 6 in the next vacant position, which occurs at position 7. Observe that we do not create a descent at the filled position to its left, but we do create a descent at position 7, since the position 8 is already filled: w "¨1 5 3 2 4 6 9 8 7

At
Step ps`1q, we can apply the previous case for i " s`1, adding the values k s`1 " n, p s`1 " 0, q s`1 "´n`1 to τ . This procedure will create descents only at those p j´1 which are still vacant.
Given a triple τ " pk, p, qq, the dual triple is defined by τ˚" pk, q, pq, where p and q were switched. Clearly, a dual triple could not be a theta-vexillary permutation, but the dual triple is useful to compute the inverse of wpτ q.
The dual triple τ˚" pk, q, pq determines a signed permutation ιpwpτ˚qq in S 2n`1 using the following algorithm: Step (0): Put a zero at the position 0; Step (1): Starting in the q 1 position, place k 1 consecutive entries, in increasing order, ending with´p 1 . Mark the absolute value of these numbers as "used" and fill the reflection through 0 with the respective reflection wpaq " wpaq; Step (i): For 1 ă i ď s, starting in the q i position (if q i ă 0 then use a position before zero), or the next available position to the right, fill the next available k i´ki´1 positions with entries chosen consecutively from the unused absolute numbers, in increasing order, ending with´p i or, if it is not available, the biggest unused number below´p i . Again, mark the absolute value of these numbers as "used" and fill the reflection through 0 with the respective reflection wpaq " wpaq; Step (s+1): Fill the remaining available positions after 0 with the unused positive numbers in increasing order. Finally, fill the reflection through 0 with the respective reflection wpaq " wpaq The difference here compared to the construction using the theta-vexillary permutation is that we allow to have negative positions, so we need the full form of the permutation. The signed permutation wpτ˚q is obtained from ιpwpτ˚qq by restricting it to the positions t1, . . . , nu.
Proof. We can prove in the same way as Lemma 2.3 of [4], adding the fact that for a ď i ď s, the permutation ιpwq maps the set apiq to bpiq and, hence, the inverse ιpwq´1 maps bpiq to apiq.
Although wpτ˚q is not theta-vexillary, a similar version of Proposition 7 holds for this case and the proof follows that same idea. Proposition 9. Let w " wpτ q be a theta-vexillary signed permutation, for a thetatriple τ " pk, p, qq. Then all the descents of w´1 are at positions q i´1 , when i ă a, and q i , when i ě a. In fact, we have w´1pq i´1 q ą p i ě w´1pq i q , for i ă a; w´1pq i q ą p i´1 ě w´1pq i`1 q , for i ě a; and there are no other descents.

Extended diagrams for theta-vexillary permutations
In this section, we aim to understand how a theta-vexillary permutation looks like in the extended diagram.
Given a position pp, qq in the extended diagram D`pwq, define the left lower region of pp, qq by the set boxes in the extended diagram strictly south and weakly west of the SE corner pq´1, pq. In other words, denoting it by Λpp, qq, this set is Λpp, qq :" tpa, bq P D`pwq | a ě q, b ď pu.
Notice that the construction algorithm of a theta-vexillary permutation wpτ q can also be seen as a process of placing dots in the extended diagram, since each pair pwpiq, iq corresponds to a dot in the diagram. We can say that a Step piq places dots in the diagram using the following rule: if an entry x is placed at a position z in the permutation, i.e., wpzq " x, then it produces a dot at the box px, zq in the diagram. For instance, the triple τ " p3 4 5 6 9, 8 6 5 4 2, 7 4 2 3 6q of Example 1 whose diagram is represented in Figure 2. The first step places the entries 9, 8 and 7, respectively, at positions 8, 9 and 10, which correspond to place dots in boxes p9, 8q, p8, 9q and p7, 10q. The second step places only a dot in the box p4, 6q. The next steps places all other dots in the diagram. Proposition 10. Let w " wpτ q be a theta-vexillary signed permutation and τ " pk, p, qq be a theta-triple. Then we have the following: (1) The boxes pq i´1 , p i q and their reflection pq i , p i´1 q are SE corners of the diagram of ιpwq (not necessary all of them); (2) For any 1 ď i ď s`1, all the dots placed by Step piq in the diagram are inside region Λpp i , q i q and outside Λpp i´1 , q i´1 q; (3) k i is the number of dots inside the region Λpp i , q i q.
Proof. Lemma 3 says that there is a symmetry between boxes pq i´1 , p i q and their reflection pq i , p i´1 q. Then, it suffices to prove that every pq i´1 , p i q is a SE corner. If p ą 0, then a signed permutation w has a descent at position p´1 if and only if ipwq has descents at position p´1 and p. By proposition 7, ιpwq satisfies ιpwqpp i´1 q ą q i ě ιpwqpp i q, and it implies that On the other hand, by Proposition 9, ιpwq´1 satisfies ιpwq´1pq i´1 q ą p i ě ιpwq´1pq i q, for any i. This proves that pq i´1 , p i q satisfies Equation (2.2), which proves item (1).
For item (2), first of all, observe that every entry x placed at position z in Step piq satisfies p i ď z and x ď q i , implying that the correspondent dot at box px, zq in the diagram belongs to Λpp i , q i q. Now, we need to check that all dots placed by Step piq are outside Λpp i´1 , q i´1 q. It is enough to verify that whenever in Step piq we are placing an entry x at a position z ě p i´1 in the permutation, then x ą q i´1 . Set β " pk i´ki´1 q´pp i´1´pi q the number of entries to be placed after the position p i´1 during Step piq. If 1 ď i ă a then condition B1 1 implies that β ă q i´1´qi . The entries that will be placed are q i`β`1 , . . . , q i and they are all strictly greater than q i´1 (in the diagram, it is equivalent to say that we have q i´1´qi available rows to place the dots above q i´1 but we only need β rows). If i " a then by Lemma 6, x ą 0 ą q i´1 .
If a ă i ď s`1 then condition B2 1 implies that β ă pq i´1´qi q´pk Rpi´iq´kRpiq q, which means that have pq i´1´qi q´pk Rpi´1q´kRpiq q available rows in the diagram to place the dots above q i´1 but we only need β rows. Observe that we must skip k Rpi´1q´kRpiq rows in the diagram since their reflection have already been used between Steps pRpiq`1q to pRpi´1qq. This proves item (2).
Finally for (3), k i is the total of dots placed until Step piq and they are all placed inside the region Λpp i , q i q. Any other dot placed after this step is placed outside Λpp i , q i q.
If we denote τ as the set tpk i , p i , q i q | 1 ď i ď su, then Proposition 10 tell us that τ as a subset of corners, i.e., we can denote τ Ă C pwq.
Remember that there is a poset "ă" in the set of corners C pwq where two corners positions satisfy pp, qq ă pp 1 , q 1 q if and only if p ą p 1 and q ă q 1 . Also remember that the NE path Nepwq Ă C pwq is the set of minimal elements of this poset. Lemma 11. Let w " wpτ q be a theta-vexillary signed permutation, and τ " pk, p, qq be a theta-triple. Then every corner position pp i , q i q of τ is minimal in the poset "ă", i.e., τ Ă Nepwq.
Proof. Suppose that there is a pair pp i , q i q of τ and a corner position pp, qq P C pwq such that pp, qq ă pp i , q i q, i.e., p ą p i and q ă q i . The pair pp, qq is not in τ because p and q are strictly decreasing s-tuples. Since the box pq´1, pq is a SE corner, Equation (2.2) implies that q ă x and p ď y, (4.1) where x :" wpp´1q and y :" w´1pqq.
When we use the construction algorithm to produce the permutation w, observe that the position p´1 must be filled by some step and the entry q must be placed in some step. So, there must be integers 1 ď m, l ď s`1 such that: a) The entry x is placed in the position p´1 during some Step pmq. This places a dot at the box px, p`1q P Λpp m , q m q; b) The entry q is placed in the position y during some Step plq. This places a dot at the box pq, yq P Λpp l , q l q. Although there exist such integers m and i, we are going to show that they cannot be either equal, smaller or greater than each other. Hence, this contradicts the assumption that pp i , q i q is not minimal in the poset.
If m " l then, using Equation (4.1), p´1 ă y are positions in Step pm " lq and the entry in such positions are wpp´1q " x ą q " wpyq, i.e., there is a descent in it. This contradicts the fact that there are no descents in a step.
If m ă l then, using Equation (4.1), we got that y ď p and q ď x ď q m (the former relation comes from the fact that every entry placed by Step pmq is weakly smaller than q m ). This implies that the box pq, yq also belongs to the region Λpp m , q m q, a contradiction of item 2 of Proposition 10.
If m ą l then observe that Step plq must fill all positions from p l to y in the construction algorithm of the permutation w. Since y ą p´1 ě p i ě p l (because i ă l), we have that the position p´1 is also filled by Step plq, which contradicts the fact that it is filled during Step pmq.
Recall that a corner position pp, qq of C pwq is unessential if there are corners pp 1 , q 1 q, pp 2 , q 2 q and pp 3 , q 3 q in the NE path Nepwq such that pp, qq is not a minimal corner in the poset in the upper half of the diagram, the box pq 1´1 , p 1 q lies above and in the same column of the box pq´1, pq, and the box pq 2 , p 2´1 q reflected from pq 2´1 , p 2 q lies to the right and in the same row of pq´1, pq, as in Figure 3.
Proposition 12. Given w P W n , suppose that the set of corner C pwq is the disjoint union C pwq " Nepwq 9 YUpwq. Then w is a theta-vexillary.
Proof. Suppose that the set of corners C pwq of a permutation w is given by the disjoint union of the NE path Nepwq and the set of unessential corners Upwq. Since all corner positions pp i , q i q of Nepwq can be ordered so that p 1 ě p 2 ě¨¨¨ě p r ą 0 and q 1 ě q 2 ě¨¨¨ě q r , set k i as the rank r w pp i , q i q and define the triple τ 1 " pk, p, qq. We will prove that τ is almost a theta-vexillary triple, i.e., it satisfies A1, A2, A3, C1, C2, and B1. In order to get B2 and B3, occasionally some elements pk i , p i , q i q should be removed from τ 1 .
Conditions A1, A2 and A3 are true because w is a signed permutation in W n . In fact, A1 and A3 come direct from the fact that there is no SE corner at row´1 or above the middle in column´1 since wp0q " 0, and A2 is satisfied just because we cannot have dots lying in opposite rows. Now, a and Rpiq, for a ď i ď s, can be defined. Let us prove that τ satisfies the remaining conditions. Consider the diagrams sketched in Figure 4.
Conditions C1 and C2 For condition C1, let a ď i ď s and consider the regions A and B as in Figure  4 (left). Denote by dpAq and dpBq the number of dots in each one of them. The definition of Rpiq can be translated to the diagram as follows: Rpiq is the unique index smaller than a such that there is no other corner of τ lying to the right of it and in the rows q Rpiq´1 , . . . , q i . Suppose that there is a dot in the darker region A of Figure 4 (left). This dot must be placed by some Step pjq, for j ą Rpiq, which implies that the corner position pp j , q j q is located above the row q i and it also places a dot above q i . However, the construction of a step says that we must fill all entries between them, including q i . So, we should have a dot at row q i and another in the row q i , a contradiction of condition A2. Hence, dpAq " 0. On the other hand, dpBq ď´q i because condition A2 says that we cannot have dots in opposite rows. Thus,´q i ě dpAq`dpBq " k i´kRpiq , since dpAq`dpBq is the amount of dots to be placed from Step pRpiq`1q to piq.
By Lemma 6, we may show that τ satisfies condition C2 1 instead of C2. In the previous case, we proved that region A contains no dots. It means that no step from pRpiq`1q to pa´1q place dots in A, which is equivalent to say that all entries placed by Steps pRpiq`1q to pa´1q are strictly bigger than q i , proving C2 1 .
For condition B1, let 1 ď i ă a´1 and consider the rectangular regions A and B as in Figure 4 (middle). Denote by dpAq and dpBq the number of dots in each one of them. Notice that the number of dots in each rectangle is limited by the length of their sides, and dpAq`dpBq is the number of dots in Step piq. If p i " p i`1 then dpBq " 0 and dpAq ă q i´qi`1 , since we cannot place a dot in row q i´1 . So, pp i´pi`1 q`pq i´qi`1 q ą dpBq`dpAq " k i`1´ki . If p i ą p i`1 then, we cannot have the dot in column p i`1 inside B because it would not create a SE corner pq i´1 , p i q. Hence, pp i´pi`1 q`pq i´qi`1 q ą dpBq`dpAq " k i`1´ki .
For conditions B2 and B3, let a ď i ď s and consider the rectangular regions A, B and C of Figure 4 (right). Suppose that p i ą p i`1 . Using the same argument of condition C1, all dots between the rows q i and q i`1 are in rectangle C and the number of dots in this region is dpCq " k Rpiq´kRpi`1q . As well as the previous case, dpBq ă p i´pi`1 and the number of dots in region A is dpAq ď pq i´qi`1 q´dpCq, since we cannot have dots in opposite rows. Therefore, pp i´pi`1 q`pq i´qi`1 q ą dpBq`dpAq`dpCq " pk i`1´ki q`pk Rpiq´kRpi`1q q.
The difficulty appears when p i " p i`1 . In this case, dpBq " 0 and dpAq ď pq i´qi`1 q´dpCq. Then, pp i´pi`1 q`pq i´qi`1 q ě dpBq`dpAq`dpCq " pk i`1´ki q`pk Rpiq´kRpi`1q q, which means that the equality can happen. So, we need to remove these elements from τ 1 where the equality holds. Denote the set of index I " I τ 1 Ă r1, ss by I " ti ě a | pp i´pi`1 q`pq i´qi`1 q " pk i`1´ki q`pk Rpiq´kRpi`1q qu Define τ the triple τ :" tpk i , p i , q i q P τ 1 | i R Iu Clearly, τ satisfies A1, A2, A3, C1, C2, and B1. Suppose that a ď i ă j are integers such that i, j R I and i`1, i`2, . . . , j´1 P I, i.e., i and j are consecutive indexes in τ . Then they satisfy pp i´pi`1 q`pq i´qi`1 q ą pk i`1´ki q`pk Rpiq´kRpi`1q q, pp i`1´pi`2 q`pq i`1´qi`2 q " pk i`2´ki`1 q`pk Rpi`1q´kRpi`2q q, pp i`2´pi`3 q`pq i`2´qi`3 q " pk i`3´ki`2 q`pk Rpi`2q´kRpi`3q q, . . . pp j´1´pj q`pq j´1´qj q " pk j´kj´1 q`pk Rpj´1q´kRpjq q. Therefore, pp i´pj q`pq i´qj q ą pk j´ki q`pk Rpiq´kRpjq q, and τ also satisfies B2 and B3.
Finally, observe that the extended diagram of wpτ q is exactly the extended diagram of w, which means that wpτ q " w. Now, we aim to proof the converse of Proposition 12.
Lemma 13. Let w " wpτ q be a theta-vexillary signed permutation, and τ " pk, p, qq be a theta-triple. Then for any 1 ď i ď s such that p i ą p i`1 , there is no corner position pp, qq different of pp i , q i q satisfying p ą p i`1 and q i ě q. In other words, pp i , q i q is the unique SE corner in the region highlighted in Figure 5. Proof. Suppose that there is pp, qq for some i such that p ą p i`1 . If p i ą p ą p i`1 then the position p´1 is a descent of w, which is impossible since all descents of w are at positions p i´1 and no one matches to p´1.
If p i " p and q ą q i then the box pq´1, p i q is a SE corner, and the dots in row q and column p i`1 are placed as in Figure 6. The dot placed in row q lies inside the region Λpp i`1 , q i`1 q, and outside Λpp i , q i q, implying that such dot is placed during Step pi`1q. i i + 1 Figure 6. Sketch of the diagram to prove Lemma 13.
Notice that the dot at column p i`1 cannot be placed during Step pi`1q because it would create a descent in Step pi`1q. Then, there is j ą i`1 such that Step pjq placed such dot. In this case, Step pi`1q should skip column p i`1 , which is impossible (by construction, this step places dots in all available columns between w´1pq and p i`1 ).
The NE path also can contain another kind of SE corner defined as follows: given a theta-vexillary signed permutation w and a theta-triple τ , a corner position pp, qq R τ is called optional if there are a ď i ď s and 1 ď j ă a such that p " p i , q i ă q " q j`1 and q i´1 ě q ą q i . In other words, pp, qq belongs to the NE path just between the corners pp i´1 , q i´1 q and pp i , q i q, and the box pq i´1 , p i q lies above and in the same column of pq´1, pq, as shown in Figure 7. Denote by Op τ pwq the set of all optional corners and observe that Op τ pwq Ă Nepwq. Observe that such box only occurs if the number of available rows between q i and q is smaller than the number of dots to be placed by Step piq, which is k i´ki´1 . In other words, we need to have enough dots to place during Step piq such that some of them are placed below the corner pp, qq. This implies that the following equation is satisfied: Thus, a triple τ 1 obtained by adding pk, p, qq to τ also gives the same permutation but it is not a theta-triple anymore.
Proposition 14. Let w be a theta-vexillary and τ be a theta-triple. Then, the set of corners is the disjoint union Proof. Denote by ιpτ q Ă C pιpwqq the set of all corner positions of τ and their reflections, i.e., ιpτ q " Propositions 7 and 9 state that all descents of w and w´1 are exclusively determined by elements in τ . More over, this assertion can be extended to the diagram Dpιpwqqq of ιpwq: all descents of ιpwq are at position p i´1 and p i , and all descents of ιpwq´1 are at position q i´1 and q i , where i ranges from 1 to s. Thus, if there is other SE corner, it should not create descents, but it must match existing descents. For instance, for τ of Example 1 and its diagram in Figure 2, observe that the corner position p2, 3q does not belong to ιpτ q but it is in the same row and column of two corner positions of ιpτ q, namely, p4, 3q and p2, 6q.
We conclude that if there exists a corner position T that does not belong to ιpτ q then there are corner positions T 1 , T 2 in ιpτ q such that T 1 is in the same column of T , and T 2 is in the same row of T . Then, we need to figure out when a combination of descents of corners T 1 and T 2 in ιpτ q a new corner.
Consider the diagram Dpιpwqq divided in quadrants as in Figure 8.

B
n · · · 1 n 1 · · · n 1 . . . Given pp, qq P ιpτ q, we say that the SE corner pq´1, pq belongs to: Consider two corner positions T 1 " pp 1 , q 1 q and T 2 " pp 2 , q 2 q in ιpτ q such that p 1 ą p 2 and q 1 ‰ q 2 , i.e., T 1 and T 2 are in different rows and columns. A cross descent is a box that lies in the same row of one of these corners (either T 1 or T 2 ) and in the same column of the remaining one. There are four types of cross descent boxes of T 1 and T 2 as shown in Figure 9. Namely, ‚ A cross descent of type α is the box pq 2´1 , p 1 q when q 1 ą q 2 ; ‚ A cross descent of type β is the box pq 1´1 , p 2 q when q 1 ą q 2 ; ‚ A cross descent of type γ is the box pq 2´1 , p 1 q when q 1 ă q 2 ; ‚ A cross descent of type δ is the box pq 1´1 , p 2 q when q 1 ă q 2 . Suppose that T 1 " pp 1 , q 1 q and T 2 " pp 2 , q 2 q are two corners of ιpτ q. Consider that T 1 lies in some quadrant X, T 2 lies in some quadrant Y, and they have cross descent box pa, bq of type ξ, where X, Y P tA, B, C, Du and ξ P tα, β, γ, δu. We say that this configuration has shape XξY. Also denote by c ξ pT 1 , T 2 q " pa, bq the respective cross descent box.
First of all, we need to figure out all possible shapes and, then, verify if the cross descent box of each shapes is an SE corner.
There are 64 different combination of shapes XξY, where X, Y P tA, B, C, Du and ξ P tα, β, γ, δu. However, not every shape is possible because τ is a theta-triple and T 1 , T 2 are chosen in ιpτ q. An example of impossible shape is AδA since, by definition, there is no i ă j where T 1 " pp i , q i q and T 2 " pp j , q j q such that q i ă q j . Thus, it remains only 24 possible shapes. We listed them in Table 1, divided in two categories: the shapes XξY where the cross descent box c ξ pT 1 , T 2 q belongs to the quadrants A or B, and the shapes XξY where c ξ pT 1 , T 2 q belongs to the quadrants C of D. Observe that if c ξ pT 1 , T 2 q belongs to quadrants C or D then its reflection c ξ pT 1 , T 2 q K belongs to quadrant A or B and corresponds to the cross descent box of corners pT 2 q K " pp 2`1 , q 2`1 q and pT 1 q K " pp 1`1 , q 1`1 q. In other words, each shape in the left column of Table 1 is equivalent to another one to the right column. Hence, we can consider only the 12 shapes where c ξ pT 1 , T 2 q belongs to quadrants A or B.
It follows from Lemma 11 that τ X Upwq " ∅ because no unessential corner is minimal in the poset. By definition of optional corner, we also have that τ X Op τ pwq " ∅ and Op τ pwq X Upwq " ∅. Then, all the sets are disjoint.
Suppose that T 1 " pp 1 , q 1 q and T 2 " pp 2 , q 2 q of ιpτ q has some shape XξY, where X, Y P tA, B, C, Du and ξ P tα, β, γ, δu, such that the cross descent box c ξ pT 1 , T 2 q is a SE corner in quadrant A or B which does not belongs to τ . Then, analyzing each situation in the first column of Table 1, we must show that either c ξ pT 1 , T 2 q P Op τ pwq 9 Y Upwq or it leads us to a contradiction. Consider ξ " α, where p 1 ą p 2 , q 1 ą q 2 , and T " pp 1 , q 2 q is a SE corner pq 2´1 , p 1 q not in τ and satisfying the following conditions ιpwqpp 1´1 q ą q 2 ě ιpwqpp 1 q ιpwq´1pq 2 q ą p 1´1 ě ιpwq´1pq 2`1 q.

(4.3)
‚ If XξY is a shape AαA, AαB or BαB, then T 1 " pp i , q i q, T 2 " pp j , q j q, where 1 ď i ă j ď s, and T " pp i , q j q is a SE corner pq j´1 , p i q. But Lemma 13 says that T cannot be a corner. ‚ If XξY is a shape AαC or BαC, then T 1 " pp i , q i q, for some i, T 2 " pp j , q j q K " pp j`1 , q j`1 q, for some j ă a, and q i ą q j`1 . We can assume that i is chosen such that there is no l ą i satisfying p i " p l and q i ą q l ą q j`1 , i.e., there is no corner of τ in the same column and between the SE corners T 1 and T . If p i ą p i`1 then Lemma 13 is contradicted. Thus, we have that p i " p i`1 and q i ą q j`1 ą q i`1 , implying that T is an optional SE corner. ‚ If XξY is a shape AαD, then T 1 " pp i , q i q, for some i ă a, T 2 " pp j , q j q K " pp j`1 , q j`1 q, for some a ď j ď s, and q i ą q j`1 . As in the previous case, we can assume that i is chosen such that there is no l ą i satisfying p i " p l and q i ą q l ą q j`1 , i.e., there is no corner of τ in the same column and between the SE corners T 1 and T . If p i ą p i`1 , then the corner T contradict Lemma 13. Thus, p i " p i`1 , q i ą q j`1 ą q i`1 and i " Rpjq.
Notice that the dot in the row q j`1 is between rows q i and q j since T is a corner, which is impossible as shown in proof of Proposition 12 (see Figure  10). Figure 10. Sketch of the diagram of shape AαD.
Consider ξ " β, where p 1 ą p 2 , q 1 ą q 2 , and T " pp 2 , q 1 q is a SE corner pq 1´1 , p 2 q is a SE corner not in τ and satisfying the following conditions ‚ If XξY is a shape AβA or AβB, then T 1 " pp i , q i q and T 2 " pp j , q j q, for some i ă a and i ă j. Observe that the dots at column p j and row q j are placed by Step pjq (or some previous one). Then, by construction, the dot at row q i´1 must be placed by some Step plq for l ď j. Thus, ιpwq´1pq i´1 q ě p j , a contradiction of Equation (4.4). ‚ If XξY is a shape BβB, then T 1 " pp i , q i q and T 2 " pp j , q j q, for some a ď i ă j ď s. If ιpwq´1pq i´1 q ă 0, i.e., the dot in the row q i´1 is in quadrant B then we can proceed as the previous case. If ιpwq´1pq i´1 q ą 0 then belongs to the quadrant C is a reflection of a dot placed during some Step plq for l ă a. Since ιpwqpq i q ă p j`1 ď 0, then q l " q i`1 and the corner pp l , q l q lies in row q i`1 . Therefore, the reflection pp l , q l q K is in the row q i´1 , and the corner T is optional or unessential (see Figure 11).
i j l l ⊥ Figure 11. Sketch of the diagram of shape BβB.
Row q j → Figure 12. Sketch of the diagram of shape AγD or BγD.
‚ If XξY is a shape BγC, then we clearly have that T is an unessential or optional corner.
Proposition 15. For w P W n , w is theta-vexillary if and only if the set of corner C pwq is the disjoint union C pwq " Nepwq 9 YUpwq.
Remark 16. If w is a theta-vexillary signed permutation but we don't know a theta-triple such that w " wpτ q, we can use the process in the proof of Proposition 12 to get τ . Basically, set τ with all the corners in the NE path Nepwq. Then, withdraw all the optional corners from it, which results in a valid theta-triple τ of w.
Proposition 17. The theta-triple is unique for each theta-vexillary signed permutation.
Proof. Suppose that τ andτ are two different theta-triples such that w " wpτ q " wpτ q. Then, τ 9 YOp τ pwq " Nepwq "τ 9 YOpτ . If there is a corner position pp, q 1 q P Op τ Xτ then there is q 2 ą q 1 such that pp, q 2 q P τ is a corner position immediately above it. Notice that pp, q 2 q does not belong toτ , otherwise condition B2 ofτ for both corners would contradict Equation (4.2) for the optional corner pp, q 1 q. Then, pp, q 2 q P Opτ X τ . For the same reason, there is q 3 ą q 2 such that pp, q 3 q P Op τ Xτ , and keep going. Hence, this process should be repeated forever, which is impossible since the sets are finite. Therefore, Op τ Xτ " H, and by the same reason Opτ X τ " H, which implies that τ "τ .

Pattern avoidance
Recall that given a signed pattern π " πp1q πp2q¨¨¨πpmq in W m , a signed permutation w contains π if there is a subsequence wpi 1 q¨¨¨wpi m q such that the signs of wpi j q and πpjq are the same for all j, and also the absolute values of the subsequence are in the same relative order as the absolute values of π. Otherwise w avoids π.
Assume that w is a theta-vexillary signed permutation. To prove that it avoids all these 13 patterns, we will assume if one of these patterns is contained in w, then show that there is a SE corner T such that T R Nepwq Y Upwq.
Suppose that w contains r2 1 4 3s as a subsequence pwpaq wpbq wpcq wpdqq satisfying wpbq ă wpaq ă wpdq ă wpcq for some a ă b ă c ă d as in Figure 13 (left). Then, there is at least one SE corner in each shaded area, which will be denoted by T and T 1 . Clearly T R Nepwq and, by Proposition 15, it should be an unessential corner. Since τ is a theta-triple of w, there are i ă j such that Step piq places the dot in the column b and Step pjq place the dot in the column a. However it lead us to a contradiction because we cannot place a dot in the row wpbq during Step piq and skip the row wpaq since it will be place further. Figure 13. Situation where w contains: r2 1 4 3s in the left; and r1 3 2s in the right. Now, suppose that w contains r1 3 2s as a subsequence pwpaq wpbq wpcqq satisfying wpaq ă wpcq ă wpbq for some a ă b ă c as in Figure 13 (right). Then, there is at least one SE corner in each shaded area, which will be denoted by T and T 1 . By definition, T is neither an unessential corner nor belongs to the NE path, i.e, T R Nepwq Y Upwq, which contradicts Proposition 15.
Notice that we could consider the diagram of w restricted to the columns a, b, c and rows 0,˘wpaq,˘wpbq,˘wpcq. Then, the corners T and T 1 in such restriction can be easily represented as the first diagram of Figure 14. Clearly, such configuration tell us that T is neither unessential nor minimal. The same idea can be used to prove that the remaining eleven patterns in Figure 14 should be avoided. Now, let us assume that w is permutation that avoids all the thirteen patterns listed above. We are going to prove that C pwq " Nepwq Y Upwq, and hence, w is a theta-vexillary permutation.
Suppose that there are corners T " pp, qq and T 1 " pp 1 , q 1 q such that q ą 0 ą q 1 and p 1 ą p ą 0, i.e., T is in quadrant A, T 1 is in quadrant B, and T 1 ă T . If we Figure 14. Diagram of w restricted to 11 different patterns. denote a :" p, b :" p 1´1 and c :" w´1pq 1 q, then they satisfy 0 ă a ă b ă c and wpaq ă 0 ă wpcq ă wpbq. Observe that a, b, c are the columns of the dots in Figure  15. In order to relate the subsequence pwpaq wpbq wpcqq of w to some 3-pattern π, we need describe all possible orderings of wpaq, wpbq and wpcq.
By hypothesis, the pattern in each case should be avoid. Hence the configuration in Figure 15 is impossible. Now, suppose that there are corners T " pp, qq and T 1 " pp 1 , q 1 q such that q ą q 1 ą 0 and p 1 ą p ą 0, i.e., both T and T 1 are in quadrant A and T 1 ă T . Denote i :" w´1pq`1q, a " p, b " p 1´1 and c " w´1pq 1 q.
If i ą 0, then they satisfy 0 ă i ă a ă b ă c and wpaq ă wpiq ă wpcq ă wpbq. Observe that ı, a, b, c are the columns of the dots in Figure 16 (left).  Figure 16. Sketch for the case where both T and T 1 are in quadrant A, and T 1 ă T .
In order to relate the subsequence pwpiq wpaq wpbq wpcqq of w to some 4-pattern π, we need to describe all possible orderings of wpiq, wpaq, wpcq and˘wpbq.
Therefore, Propositions 15 and 18 prove Theorem 1. This pattern avoidance criterion allow us to compare a theta-vexillary permutation with a vexillary permutation of type B defined by Billey and Lam [5]. According to them, a signed permutation is vexillary of type B if the Stanley symmetric function F w is equals to the Schur Q-function Q λ for some shape λ $ ℓpwq with distinct parts. In [5,Theorem 7], they proved that a vexillary permutation of type B should avoid eighteen patterns, which includes all thirteen patterns given in our main theorem. Then, every vexillary permutation of type B is theta-vexillary.
Therefore, for an arbitrary theta-vexillary signed permutation w, we cannot say that the Stanley symmetric function F w is equals to the Schur Q-function of w. In principle, this property is not expected since the equality holds for every vexillary signed permutation as proved by [4,Corollary 5.2].