A geometric and combinatorial exploration of Hochschild lattices

Hochschild lattices are specific intervals in the dexter meet-semilattices recently introduced by Chapoton. A natural geometric realization of these lattices leads to some cell complexes introduced by Saneblidze, called the Hochschild polytopes. We obtain several geometrical properties of the Hochschild lattices, namely we give cubic realizations, establish that these lattices are EL-shellable, and show that they are constructible by interval doubling. We also prove several combinatorial properties as the enumeration of their $k$-chains and compute their degree polynomials.

and cell complexes introduced by Saneblidze [San09,San11] in the area of algebraic topology. These cell complexes are called Hochschild polytopes by Saneblidze. They provide, in the context of algebraic topology, combinatorial cellular models of free loops spaces. There are several ways to build Hochschild polytopes. For instance, they can be obtained by a sequence of truncations of the -simplex, where is the dimension of the polytopes [RS18].
It is shown in [Cha20] that the set of Dyck paths in these specific intervals in dexter posets is in bijection with a set of words defined on the alphabet {0 1 2} satisfying some conditions. Better than that, by considering the poset on this set of words endowed with the componentwise order, Chapoton shows that a covering relation on Dyck paths for the dexter order implies by this bijection a covering relation on the corresponding words.
As a first contribution of the present work, we show the reverse implication. This implies that the two posets are isomorphic. Moreover, we show that these posets are lattices. Because of their links with cell complexes of Saneblidze, we call these lattices Hochschild lattices. Our goal is to present a geometric and combinatorial exploration of Hochschild lattices, revealing several interesting features. To this aim, we shall mainly work with the word version of the lattice previously mentioned, whose elements are called triwords.
In the first section, we recall several definitions by starting with the one of the dexter semilattices and the bijection between Dyck paths of the specific intervals and triwords. We divide our study of the posets into two strands: a geometric one and a combinatorial one. Section 2 is devoted to the geometric properties. First, we provide a natural geometric realization for Hochschild lattices, by placing triwords of size in the space R and by linking by an edge triwords which are in a covering relation. Thanks to this realization, called cubic realization, we are able to show that Hochschild lattices are EL-shellable and constructible by interval doubling (or equivalently congruence uniform [Müh19]). Section 3 is about enumerative and combinatorial results. We give here for instance the degree polynomial of the Hochschild lattices that enumerates the triwords with respect to their coverings and the elements they cover. We also provide a formula to compute the number of intervals of these lattices, as well as a method to compute the number ofchains. Section 3 ends with the introduction of an interesting subposet of the Hochschild poset, which seems to have similar nice properties. An appendix on Coxeter polynomials written by Chapoton is added at the end of this article. Acknowledgements. The author was supported by the ANR project Combiné ANR-19-CE48-0011. The author would also like to thank Frédéric Chapoton for the subject and for his many advices.
1. D 1.1. Hochschild polytopes and triwords. Let 0 and = 1 2 be a word of size . The prefixes of are the + 1 words , 1 , and the suffixes of are the + 1 words , , with ∈ [ ]. A word is a factor of if there is a prefix and a suffix such that = . A word is a subword of if can be obtained by deleting letters in . For instance, is a subword of .
To describe our objects introduced in the sequel, we use the regular expression notation [Sak09]. Recall that for a letter , * denote the set of words for any ∈ N, and + denote the set of words * . Besides, for two expressions and , + is the union of the two sets denoted by and .
For any 0, a Dyck path of size is a lattice path from (0 0) to (2 0) which stays above the horizontal line, and which consists only of north-east steps and southeast steps. The graded set of Dyck paths is denoted by Dy where the size of a Dyck path is its number of north-east steps. To simplify, we see a Dyck path of size as a binary sequence of length 2 where the letter 1 encodes a north-east step and the letter 0 encodes a south-east step. In this section, we recall several definitions, concepts, and notations given in [Cha20].
Let ∈ Dy( ). The Dyck path is primitive if for all Dyck paths and such that = , one has = or = . A factor is a subpath of if is a Dyck path. A subpath of is movable if is primitive and if there is a prefix and a suffix such that = 10 , where > 0, and either = or the first letter of is 1. Figure 1 The set Dy( ) endowed with the dexter order is a meet-semilattice with many properties highlighted in the article of Chapoton. In this article, we restrict ourselves to a particular interval of this semilattice.
In any Dyck path , a factor 01 is called a valley. The height of a valley is the ordinate of its corresponding middle point in the path.
For any 1, let F( ) be the interval in Dy( + 2) between 1100(10) and 11 0 100. In particular, any in the interval F( ) satisfies the three following assertions: ⋆ the sequence of heights of the valleys in is weakly decreasing from left to right, ⋆ the Dyck path ends either with 010 or 0100, ⋆ the Dyck path starts with 11 and has only valleys of height 0 or 1.

For any
1, let us recall the bijection ρ between F( ) and the set of words of length in the alphabet {0 1 2} satisfying some conditions. Let ∈ F( ) and N 2 be an integer initially set to 0. By reading from left to right the word , let us build the word , initially the empty word, by following the two conditions, (i) when two consecutive 1 are read in , except the first two letters of , then 1 is added to N 2 , (ii) when a valley of height is read in , the word 2 N 2 is added at the end of the building word , and N 2 is then set back to 0 a .
The result ρ( ) is the word obtained after reading all . The length of is because, except the two initial letters 1, every letter 1 in contributes a letter in .
For instance, the image by ρ of the two Dyck paths 1101001010 and 1110010010, both in F(3), are respectively 100 and 120.
Since we are going to work in this article on the set ρ(F( )), we need to give a description of this set which is independent of the construction induced by ρ.
Proof. Let ∈ F( ) such that ρ( ) := . Then the first letter of is either 0 or 1. Besides, a letter 0 cannot be follows by a letter 1 because the height of the valleys in is weakly decreasing from left to right. Thus, one has ∈ Tr( ).
Moreover, we know from [Cha20] that the number of elements in F( ) is (1.9).
1.2. Isomorphism of posets. We endow the set of triwords with the componentwise order and show that the bijection ρ is an isomorphism of posets. Then, we describe the meet and join of the poset so defined.
For any 1, let be the partial order on Tr( ) satisfying for any ∈ Tr( ) such that for all ∈ [ ]. The set Tr( ) endowed with is the Hochschild poset of order .
We set that ⋖ if and only if and there is only one index such that < , and if there is ∈ Tr( ) such that , then either = or = . Obviously, the binary relation ⋖ is contained in the covering relation of (Tr( ) ).
Note that the minimal element of Tr( ) is 0 and the maximal element is 12 −1 .

Proposition 1.3. For any
1, the binary relation ⋖ is the covering relation of the Hochschild poset Tr( ).

Proof. Let
∈ Tr( ) such that covers . The case = 1 is clear. Let > 1 and let be the minimal index such that = , and let := 1 −1 +1 be the word with the same letters as , except for the -th letter. Since > , either is obtained by replacing in the -th letter 0 by 1 or by 2, or by replacing in the -th letter 1 by 2. In both cases, is not 0. Moreover, since is the minimal index such that = , if there is a letter 0 before in , then this letter exist also in , and so cannot be 1. Therefore, the subword 01 cannot be generated in . Thus, the word is a triword. It follows that there is a triword ′ such that is covered by ′ . One can conclude that between two triwords in covering relation, there is exactly one different letter.
For any Dyck path = 10 with > 0, a prefix, a suffix, and a movable subpath, let N( ) be the number of consecutive 0 letters that appear before in .
Proposition 1.4. For any 1, the map ρ is an isomorphism of posets from F( ) to Tr( ).

Let
∈ Tr( ) such that ⋖ , and let and be the respective images of and by ρ −1 . Since ⋖ , there is only one index such that < . Then, there are three cases: either 0 becomes 1 or 0 becomes 2, or 1 becomes 2.
⋆ Suppose that = 0 and = 1. Then, in the path , there is a movable subpath (in blue (dark) in (1.10)) starting at the height 0 such that N( ) 2. The height of the starting point of gives the value of in by the map ρ. In the path , since only one letter changes between and , the same subpath starts at the height 1 and N( ) = N( ) − 1. Because of this move, we have to add one 0 after .
(1.10) ⋆ Suppose that = 0 and = 2. Then, in the path , there is a movable subpath (in blue (dark) in (1.11)) starting at the height 0, followed by an other subpath also starting at the height 0. This is the height of the starting point of which gives in by the map ρ. In the path , there is a subpath starting at the height 0 followed by the subpath which is unchanged, such that N( ) = 0 and N( (1.11) ⋆ Suppose that = 1 and = 2. This case is very similar to the previous case, by changing the height of the starting point 0 of , and by 1.
In all cases, one has Dex .
Let us describe the join and the meet between two triwords and .
Let ∈ Tr( ), and let := max( 1 1 ) max( ). Since 1 and 1 are both none 2, 1 = 2. Besides, if = 0 for ∈ [ ], then necessarily and have to be equal to 0. In this case, for all > , neither nor can take the value 1. Therefore, if there is an index ∈ [ ] such that = 0, then = 1 for all > . Thus is a triword.
The triword is the join between and . Indeed, is by definition the smallest element such that for all ∈ [ ], and . Moreover, since the join between and is unique, by Proposition 1.4, the Hochschild poset is a join-semilattice. One can conclude that Hochschild poset is a lattice since there is a unique minimal triword [Sta11]  Proof. If := min( 1 1 ) min( ) is a triword, then = . Suppose that is not a triword. Since we replace in all subwords 01 by 00, is a triword. Moreover, if there is a subword 01 in , then either or has a letter 0 following by letters 0 or 2. Necessary, the word inherits this letter 0, and then is a triword if all letters after this letter 0 are 0 or 2. Therefore, the triword is the greatest element such that and .
For example, in order to compute 11112 ∧ 10222, first we compute = 10112, which is not a triword. We replace the subword 2 3 and 2 4 by the subword 00. One has 11112 ∧ 10222 = 10002.

G
Through triwords, it is possible to give a cubic realization of the Hochschild lattice by placing in the space R all triwords of size . This lattice thus joins the family of posets having a cubic realisation [Com19,CG20]. This realization allows us to show two geometrical results: on the one hand that the Hochschild lattice is EL-shellable and on the other hand that this lattice is constructible by interval doubling.
2.1. Cubic realizations. The Hochschild poset Tr( ) can be seen as a geometric object in the space R by placing for each ∈ Tr( ) a vertex of coordinates ( 1 ), and by forming for each ∈ Tr( ) such that ⋖ an edge between and . We call cubic realization of Tr( ) the geometric object C (Tr( )) just defined. Figure 2.1 shows the cubic realization of the poset Tr(2) and the poset Tr(3).
The first thought that comes to mind, is that for any 1, any -face of the realization C (Tr( )) is contained in a − 1-face of the hypercube of dimension , for ∈ [0 − 1].  Indeed, between the minimal triword 0 := and the maximal triword 12 −1 := , there is no triword of size such that < < for all ∈ [ ] since 1 = 0 and 1 = 1.
Therefore, we can see this realization as one empty cell of dimension . Thus, it is clear that the volume of C (Tr( )) is 2 −1 .

EL-shellability.
In [BW96] and [BW97], Björner and Wachs generalized the method of labellings of the cover relations of graded posets to the case of non-graded posets. In particular, they showed the EL-shellability of the Tamari poset [BW97]. In this section, we show that the Hochschild lattice is EL-shellable.
A poset is bounded if it has a unique maximal element and a unique minimal element for . A chain in is a sequence of elements Let ⋖ be the covering relation of . If ⋖ +1 for all ∈ [ − 1], then the chain (2.1) is saturated.
By a slight abuse of notation, the set of elements ( ) such that ⋖ is also denoted by ⋖ . Let be a bounded poset and Λ be a poset, and λ : ⋖ → Λ be a map. For any saturated chain (1) ( ) of , we set We say that a saturated chain of is λ-increasing (resp. λ-weakly decreasing) if its image by λ is an increasing (resp. weakly decreasing) word for the order relation Λ . We say also that a saturated chain (1) ( ) of is λ-smaller than a saturated chain (1) ( ) of if λ (1) ( ) is smaller than λ (1) ( ) for the lexicographic order induced by Λ . The map λ is called EL-labeling (edge lexicographic labeling) of if for any ∈ satisfying , there is exactly one λ-increasing saturated chain from to , and this chain is λ-minimal among all saturated chains from to . Any bounded poset that admits an EL-labeling is EL-shellable (see [BW96,BW97]).
The EL-shellability of a poset implies several topological and order theoretical properties of the associated order complex ∆( ) built from . Recall that the faces of this simplicial complex are all the chains of . For instance, if has at most one λ-weakly decreasing chain between any pair of elements then the Möbius function of takes values in {−1 0 1}. In this case, the simplicial complex associated with each open interval of is either contractible or has the homotopy type of a sphere [BW97].
In order to show the EL-shellability of Tr( ) for 1, we set Λ as the poset Z 2 ordered lexicographically. Then we introduce the map λ : ⋖ → Z 2 defined for any such that ⋖ by λ( where is the unique index such that = . Observe that because of the covering relation ⋖ defined in Proposition 1.3, the image by λ of any saturated chain in Tr( ) is well-defined. Theorem 2.1. For any 1, the map λ is an EL-labelling of the Hochschild lattice Tr( ). Moreover, there is at most one λ-weakly decreasing chain between any pair of comparable elements of Tr( ).

Proof. Let
∈ Tr( ) such that and which is not necessarily saturated. Then, by concatenating the unique saturated chain in each interval [ ( −1) ( ) ] for all ∈ [ ], we obtain a saturated chain between and .
Since each word ( ) of this saturated chain is obtained from by replacing letters from left to right, this chain is clearly weakly increasing for the partial order . Furthermore, between two consecutive triwords ( −1) and ( ) in this saturated chain, ( −1) ⋖ ( ) . Therefore, the image of the chain by λ is increasing for . Thus this chain is λ-increasing.
Moreover, since between any two consecutive triwords of this chain only one letter is different, if we consider another saturated chain from to , then at some point, this chain passes through a word obtained by increasing a letter which has not the smallest possible index. It lead us to choose later in this chain the letter with a smallest index to increase it. For this reason, the saturated chain obtained is not λ-increasing.
If a λ-weakly decreasing chain exists in [ ], then it must have the sequence of edgelabels (( Indeed, suppose that between and , there is an index ∈ D( ) such that = 0 and = 2, and there is a triword such that with = 1. Then, for this index , the sequence of edge-labels passing through is (( 0) ( 1)), and so the saturated chain passing through in [ ] cannot be λ-weakly decreasing. Therefore, to obtain a λ-weakly decreasing chain in [ ], each index of D( ) can only appear once in the sequence of edge-labels.
Assume that there is a λ-weakly decreasing chain. For the same reason as previously, this chain is unique. having ′ as order relation, which is defined as follows. For any ∈ [I], one has ′ if one of the following assertions is satisfied: This operation has been introduced by Alan Day as an operation on posets preserving the property of being lattices. A lattice is constructible by interval doubling (bounded in the original article) if is isomorphic as a poset to a poset obtained by performing a sequence of interval doubling from the singleton lattice.
For all 1, let us build Tr( + 1) from Tr( ) by following these three steps.
(i) Let T 0 ( + 1) be the poset on the set of all words 0 such that ∈ Tr( ).
Proof. Let ∈ T( + 1), is written either 0, or 2 with ∈ Tr, or is a word of form 1(1 + 2) * 1. It is clear that, for any ∈ Tr( ), adding a letter 0 or a letter 2 at the end of give a triword of size + 1. Likewise, a word of form 1(1 + 2) * 1 is also a triword. Now, let ∈ Tr( + 1). Suppose that +1 = 1. Since the subword 01 is forbidden, one has ∈ {1 2} for all ∈ [ ]. Therefore, belongs to T( + 1). Suppose that +1 = 0 or that +1 = 2. Since belongs to Tr( + 1), the conditions of triwords remain on the prefix of size of . Thus, one has ∈ Tr( ). Proof. We proceed by induction on 1. If = 1, we have the poset 2, namely the poset with two elements, which is a lattice constructible by interval doubling. Assume now that 2. We have to show that Tr( +1) can be obtained from Tr( ) by a sequence of interval doublings. By Lemma 2.2, one has that Tr( + 1) is the poset T( + 1). Since T( + 1) is obtain from Tr( ) by performing the three steps (i), (ii), and (iii), by showing that these two last steps are two operations of interval doubling, the intended result will follow.
In the step (iii) one builds I 1 from I 0 by changing for all ∈ I 0 the letter 0 to 1. Since for all ∈ I 0 such that , any word such that is by definition of a word of shape 1(1 + 2) * 0, one has that I 0 is the interval [1 0 12 −1 0]. For the same reason, I 1 is the interval [1 +1 12 −1 1].

C
In this section, several combinatorial and enumerative properties of the Hochschild lattice are proved. We obtain results such as the enumeration of intervals, the enumeration of -chains, and the description of the degree polynomial of the Hochschild lattice.
3.1. Irreducible elements and maximal chains. Here we give some general properties of the Hochschild lattice, such as its degree polynomial and a description of joinirreducible and meet-irreducible elements.
Recall that an element of a lattice is join-irreducible (resp. meet-irreducible) if covers (resp. is covered by) exactly one element in . We denote by J( ) (resp. M( )) the set of join-irreducible (resp. meet-irreducible) elements of . Moreover, let be a saturated chain of where ⋖ is the covering relation of . The length of the saturated chain (3.1) is − 1 c . A longest saturated chain between the minimal element and the maximal element of is a maximal saturated chain.
Let us describe the set of join-irreducible and meet-irreducible elements of Tr( ) by using the regular expression notation [Sak09] recalled in Section 1.1.
The two possibilities of having a join-irreducible triword are either to change a letter = 1 to 0 such that all letters on the left of are letters 1 and letters on the right of are 0, or to change a letter = 2 to 0 such that all other letters are 0. Indeed, suppose that we change in a triword a letter = 2 to 1. Since should cover just one triword, all other letters in have to be 0. However, since the first letter in is different from 2, there is a letter −1 such that −1 = 0. Thus, −1 can be also decreased. This implies c In Section 3.3, we deal with -chains, where refers not to the length of the chain but to the number of elements forming that chain. that covers more than just one triword. Since the subword 01 is not allowed, the set of triwords which covers a unique triword is described by

J(Tr( )) = { ∈ Tr( ) :
∈ 1 + 0 * + 0 + 20 * } (3.2) Likewise, the three possibilities of having a meet-irreducible triword are either to change a letter 1 to 2 or to change a letter 0 to 1, or to change a letter 0 to 2. Moreover, for all cases, the other letters which are unchanged should be as large as possible. Thus, the set of triwords covered by a unique triword is described by Recall that a lattice is join-semidistributive if for all In Section 2, we have shown that the Hochschild lattice is constructible by interval doubling. However, it is known from [Day79] that lattices constructible by interval doubling are in particular semidistributive. Moreover, a finite lattice is constructible by interval doubling if and only if it is congruence uniform [Day79]. In particular, the number of join-irreducible elements J( ) is equal to the number of doubling steps needed to build [Müh19].
Therefore, there are two consequences of Theorem 2.3. The first one is that for any 1, the Hochschild poset Tr( ) is semidistributive. Another consequence is that the difference of numbers of join-irreducible elements between Tr( − 1) and Tr( ) is always 2. Indeed, Tr( ) is constructible by interval doubling from Tr( − 1) with only two steps. Proof. If = 1, then the length of the saturated chain [0 1] is 1. Suppose that > 1. Since all letters 0, except the first one, can be increased to 1, then to 2, the length of a maximal saturated chain in Tr( ) between 0 and 12 −1 is at most 2 − 1. Therefore, to obtain a maximal saturated chain between 0 and 12 −1 , all letters 0 in 0 must become 1 before becoming 2, except for the first 0. Considering that, the letters have to be increased from left to right, in order to avoid the forbidden subword 01. This way, each letter of 0 , except the first one, contributes 2 in the length of the saturated chain between the minimal triword and the maximal triword. Since the first 0 contributes 1, the length of such a saturated chain is 2 − 1.
Furthermore, since the letters have to be increased from left to right, this implies that a triword belongs to a maximal saturated chain if and only if for any letter = 0 then = 0 for all .
Let be a lattice such that the length of a maximal saturated chain is . If #J( ) = #M( ) = then is an extremal lattice [Mar92]. By Lemma 3.1 and the generating function (3.4), one has the following result. It is shown in [TW19] that if a lattice is extremal and semidistributive, then it is also left modular, and therefore trim. By Theorem 2.3, one has that Tr( ) is semidistributive, thus Tr( ) is trim.

Let
be an extremal lattice. The union of maximal saturated chains of is known as the spine of . It is known from [Tho06] that the spine of an extremal lattice is a distributive sublattice of . The spine of is denoted by S( ). Figure 3.1 shows the spine of S(Tr(2)) and S(Tr(3)).  Birkhoff states that any finite distributive lattice is isomorphic to the lattice J( ) of the order ideals of the subposet of restricted to its join-irreducible elements, ordered by inclusion [Bir37,Sta11].
Let us consider the subposet J(S(Tr( ))) of S(Tr( )). Since the spine of Tr( ) is a distributive sublattice of Tr( ), then by the FTFDL one has that S(Tr( )) is isomorphic to J (J(S(Tr( )))).
For instance, Figure 3.2 depicts the construction of J (J(S (Tr(3)))), which is a distributive lattice isomorphic to S(Tr(3)) (see Figure 3.1).  Our aim is to give a description of triwords belonging to the spine of the Hochschild lattice. Then, in this set, we give a description of join-irreducible triwords.
By Lemma 3.1 we know that a triword belongs to a maximal saturated chain if and only if for any letter = 0 then = 0 for all . Therefore, the regular expression of these triwords is Therefore, the generating function is and thus #S(Tr( )) = 2 (3.9) Let ∈ S(Tr( )). The two possibilities for to be a join-irreducible triword are either to have one unique letter 1 which can be changed to 0 or to have one unique letter 2 which can be changed to 1. To summarize, J(S(Tr( ))) = { ∈ S(Tr( )) : ∈ 1 + 0 * + 1 + 20 * } (3.10) One can deduce the generating function G J(S(Tr)) = + 2 (1 − ) 2 (3.11) and thus #J(S(Tr( ))) = 2 − 1 (3.12) From (3.10) one can also deduce that the shape of J(S(Tr( ))) is as depicted in Figure 3.3. where for any ∈ , in ( ) (resp. out ( )) is the number of elements covered by (resp. covering) in . The specialization d (1 ) is the -polynomial of .
Besides, for any letter of with ∈ [ ], the number of letters ′ such that the word ′ defined by ′ := for all = is in covering relation with is the degree of the letter . The sum of the degrees of all letters of is the number of elements covered by or covering , namely in ( ) + out ( ). Let us consider the grammar of Tr given by Lemma 1.1. By the map | | out Tr ( ) one obtains the system of formal series Indeed, in (1.2) of the grammar, 0A becomes P A ( ) because the letter 0 can always be increased to 2. Note that the letter 0 in 0A cannot be increased to 1 because in (1.4), this expression 0A comes after a first letter 0, and the subword 01 is prohibited by definition of triwords. However, 2A becomes P A ( ) since the letter 2 cannot be increased. Likewise, in (1.3), 0A becomes P A ( ) because the letter 0 can be increased to 1, and 1B becomes P B ( ) because the letter 1 can be increased to 2, unlike the letter 2 in 2B which becomes P B ( ). Thus, From this expression of P Tr ( ) in partial fraction decomposition, we deduce by a straightforward computation the given expression for d Tr( ) (1 ).
Proof. Suppose that the first letter of is 0, then all letters of are either 0 or 2. The letter 1 can be increased to 1, but since we cannot have a letter 0 followed by 1, all other letters 0 can only be increased to 2, and all letters 2 can only be decreased to 0. And so for the case where 1 = 0, all letters of have degree 1.
Suppose now that the first letter of is 1. Either 1 is the only letter 1 in or there is another letter = 1 such that all letters after are not 1. In the first case, 1 can be decreased to 0, thus all letters of have degree 1. In the second case, since there is at least one other letter 1 in , 1 cannot be decreased to 0. Then the degree of 1 is 0. However, this degree is compensated by the degree of the letter . Indeed, the last letter 1 is the only one which can be decreased to 0 or increased to 2. Hence the degree of is 2, and since all other letters of have degree 1, the sum is equal to .

Intervals and -chains.
This section also provides enumerative results about the Hochschild lattice. We have already computed the length of any maximal chain for this lattice in Section 2. Here we give a method to find formulas for the number of -chains of this lattice.
We use for a letter and a word the notation ∈ if there is a letter = . Conversely, / ∈ if all letters of are different from . Thereafter, we denote by ( ) the set of -chains of Tr( ) that contains exactly words such that 0 / ∈ . We denote by z ( ) the cardinality of ( ). Note that for = 1, z (1 ) = 1 for all First, we need to define a classification for all -chains of size .
This classification is called the -classification for -chains. Note that the union of all these sets is disjoint and give a description of all -chains.
For any 2, 1, ∈ [0 ], and , let where γ ′ is the -chain obtained by forgetting the last letter of each word of γ, and t is the number of words ending by 2 in γ.
In the two cases, the position of the first letter 2 depends on the integer t.
Thus, for γ a -chain in ( ), it follows that (2) · · · ( ) . This implies that this -chain is a -chain of Tr( ). Moreover, since the letter 0 is added at the end of ( ) for ∈ [ − ], the -chain δ belongs to ( ).
In both cases, since δ belongs to ( ), this implies that the map φ ( ) is surjective.
Let (t 1 γ ′ ) and (t 2 δ ′ ) be two pairs with t 1 t 2 ∈ [0 ], and γ ′ ∈ 1 ( − 1 ) and δ ′ ∈ 2 ( − 1 ) with 1 2 ∈ [ ]. Let γ be the image of (t 1 γ ′ ) and δ be the image of (t 2 δ ′ ) by φ . Suppose that (t 1 γ ′ ) = (t 2 δ ′ ). This implies that either t 1 = t 2 or γ ′ = δ ′ . In the first case, if t 1 > t 2 then there are more words ending by 2 in γ than in δ. Thus one has γ = δ. In the second case, there is at least one word in γ such that the prefix of this word is different from the word with the same index in δ. Here again, one has γ = δ. Hence, the map φ ( ) is injective.
For instance, for the 4-chain (3.23), γ ′ belongs to 1 (4 4) and is 2. We can rebuild γ by adding the letter 2 on the two last words of γ ′ , since by definition of triwords, the greater triwords of a -chain must have greater or equal letters compare to smaller triwords. Besides, since the two first words of γ ′ have the letter 0, we can only add the letter 0 at its end. Here γ ′ belongs to 3 (4 5). Since γ ∈ 2 (5 5), to rebuild γ from γ ′ , we have to add 0 at the end of the third word of γ ′ . Moreover, since t = 1, the letter 2 is added to the last word and the letter 1 is added to the penultimate word of γ ′ .
For any ( ) of this classification, one obtains by denoting by z ( ) the cardinality of ( ) with ∈ [0 ], the following result. The system . . .
is called z-system. where P ( ) is a monic polynomial of degree determined by the z-system.
Proof. Since for = 1, all z (1 ) = 1 with ∈ [0 ], one can rewrite the z-system with matrices  where Q ( ) is clearly polynomial in . It only remains to deduce the polynomial P ( ) from the matrix Q ( ), as the sum of all entries of Q ( ).
Furthermore, P ( ) is a polynomial of degree since appears in Moreover, a particular case from Lemma 3.8 gives that N (1 + 1) = !. Since N is a strictly upper triangular matrix, N (1 + 1) is the only nonzero entry of N . This implies that P ( ) is a monic polynomial. Note that since for = 1, all z (1 ) = 1 with ∈ [0 ], the number of -chains is + 1 for all 1. Using Proposition 3.7, one can therefore deduce that P (1) = ( + 1) +1 .
Recall that the triwords of size are enumerated by A demonstration of this result is given in Section 1.1, involving generating series. By Proposition 3.7, one has   z 0 ( 1) which leads to the formula already known, for 1, z 0 ( 1) + z 1 ( 1) = 2 −2 ( + 3) (3.37) Likewise, to enumerate the intervals of the Hochschild lattice, or in other words their 2-chains, one has The number of intervals of Tr( ) is therefore given by z 0 ( 2) + z 1 ( 2) + z 2 ( 2) = 3 −3 2 + 9 + 17 (3.39) In the same way, the number of 3-chains is The subposet (Tr µ ( ) ) is called mini-Hochschild poset. As for Hochschild posets, we can give the -classification for -chains of mini-Hochschild posets. This classification is identical to the classification (3.19). For any 2, 1, and ∈ [0 ], let us show that the map φ The z-system for the mini-Hochschild poset holds, and one has for any 2, 1, and for all ∈ [0 ], Since z (1 ) = 1 and z (1 ) = 0 for all ∈ [0 − 1], it follows that the z-system for the mini-Hochschild poset can be rewritten Thus, for any 2 and 1, the number of -chains in the poset Tr µ ( ) is given by the sum of the last column of M −1 , where M is the upper triangular matrix. One can conclude that Proposition 3.7 holds for the mini-Hochschild poset.
In the same way, from (3.38) one deduce that the number of intervals of Tr µ ( ) is Similarly to the remark on the sequence of constant terms (3.43), it seems that the sequence of constant terms of these polynomials 1 2 6 24 120 (3.54) is the sequence of factorial numbers.
Several other properties verified by the Hochschild poset seem to hold for the mini-Hochschild poset. It may be interesting to proceed to a complete study of this subposet as well. , where the second factor is the transpose of the inverse. This definition may look strange, but is very natural from a representation-theoretic point of view, where it comes from the Auslander-Reiten translation functor τ on the derived category of modules over the incidence algebra of . To keep it short, let us just say that the Coxeter matrix C , up to change of basis over Z, is an invariant of that depends only on the derived category . It is known that non-isomorphic posets can have equivalent derived categories, in which case they will share the same Coxeter matrix up to change of basis.
The Coxeter polynomial of , defined as the characteristic polynomial of the Coxeter matrix C , is therefore also an invariant of depending only on the derived category . This invariant is very easily computed on examples and sometimes turns out to have nice properties.
In the case of Hochschild posets, computer experiments suggests the following conjecture.
Conjecture 3.9. The Coxeter polynomial ( ) of the Hochschild poset Tr( ) is a product of cyclotomic polynomials.
One can note that the Coxeter matrices for Tr(4) and Tr(5) are not diagonalizable over the complex numbers and do not have finite multiplicative order.
Moreover, one can propose a guess for the factorization, as follows. Let be the Coxeter polynomial ( ) of the Hochschild poset Tr( ) if is odd and (−1) deg (− ) if is even. where the integers ( ) have the description given below.
Note that the description is to be taken as a first approximation only, as there are still ambiguities in the proposal for some exponents. This finishes the proposed description for the exponents ( ). The last case is the ambiguous place, as the known values were not sufficient to make a better guess for splitting I (D − ⌊D ⌋) into the product of an index and an exponent.