Statistics on partitions arising from seaweed algebras

Using the index theory of seaweed algebras, we explore various new partition statistics. We find relations to some well-known families of partitions as well as a surprising periodicity result. Mathematics Subject Classifications: 05A17, 17B99

of an index-based statistic is of double-edged utility, yielding insights into two ostensibly unrelated theories: the classical study of bijections between restricted partitions and the more topical study of the enumeration of certain important classes of seaweed algebras.
In the first case, (λ, 1 w(λ) ), we find a connection to classical partition theory. If P(n) is the set of partitions of weight n, then the sequence {c i n } ∞ n=1 defined by c i n = |{λ ∈ P(n) : ind 1 w(λ) (λ) = n − i}|, for each fixed i is eventually constant -stabilizing at the number of two-colored partitions of i − 1 (see Theorem 11).
In the second case, we consider seaweeds defined by the pair (λ, w(λ)). The enumeration of such composition pairs, when the index is zero, is of concern to Lie theorists. 1 Recent efforts to enumerate pairs of compositions that define a Frobenius (index zero) seaweed have concentrated on limiting the number of parts in the compositions. For example, Duflo and Yu use certain index-preserving operators on the set of compositions corresponding to a Frobenius seaweed subalgebra of sl(n) to show that if t is the number of parts in the defining compositions, then the number of such compositions is a rational polynomial of degree t 2 evaluated at n (see [17], Theorem 1.1 (b)). Duflo and Yu's result is existential in nature. However, if compositions are restricted to partitions and a modest limit is placed on the size of the parts -rather than the number of parts -the number of such compositions corresponding to a Frobenius seaweed subalgebra of sl(n) becomes a periodic function of n (see Theorem 16).
The organization of the paper is as follows. In Section 2 we develop the definitions and notation for partitions and seaweeds. In Section 3 we use the index theory of seaweeds to define index-based statistics on partitions and use this new construction to connect to some well-known classical investigations. We conclude with some open questions and directions for further study.

Preliminaries
In Section 2.1 we review standard combinatorial notation. In Section 2.2 we detail the recent index theory of seaweed algebras. Throughout this article, we tacitly assume that all Lie algebras are over the complex numbers.

Partitions
We follow the notation of Andrews [1] and adopt the following conventions.

Definition 1.
A partition λ of a positive integer n is a finite non-increasing sequence of positive integers λ 1 , λ 2 , . . . , λ m such that n = m i=1 λ i . The λ i are called the parts of the partition and w(λ) = n is the weight of the partition.
We will often employ the vector notation for the partition λ = (λ 1 , λ 2 , . . . , λ m ). It will sometimes be useful to use a frequency notation that makes explicit the number of times a particular integer occurs as a part of a partition. So, if λ = (λ 1 , λ 2 , . . . , λ m ), we alternatively write where exactly f i of the λ j are equal to i. A graphical representation of a partition, called a Ferrers diagram, is helpful to develop the notion of the conjugate of a partition. More formally, the Ferrers diagram of a partition λ = (λ 1 , . . . , λ m ) is a coordinatized set of unit squares in the plane such that the lower left corner of each square will have integer coordinates (i, j) such that The conjugate of a partition λ is the partition λ C resulting from exchanging the rows and columns in the Ferrers diagram associated to λ. The Ferrers diagram of the partition (4, 2, 1), as well as its conjugate (3,2,1,1), are illustrated in Figure 1.

Seaweed algebras
In this section, we introduce seaweed algebras in sl(n) -the set of all n × n matrices of trace zero. As we will see, such seaweed algebras are naturally defined in terms of two compositions of the positive integer n. Recall that a composition of n is an unordered partition, which we will denote by λ 1 |λ 2 | · · · |λ m to distinguish it from the ordered case in Definition 1, where there is an order relation on the λ i s.
The subalgebra of sl(n) preserving these flags is called a seaweed Lie algebra, or simply seaweed, and is denoted by the symbol a 1 | · · · |a m b 1 | · · · |b t , which we refer to as the type of the seaweed. If b 1 = n, the seaweed is called maximal parabolic.
the electronic journal of combinatorics 27(3) (2020), #P3.1 Remark 3. The preservation of flags in Definition 2 insures that seaweeds are closed under matrix multiplication, and therefore define an associative algebra, hence also a Lie algebra under the commutator bracket.
The evocative "seaweed" is descriptive of the shape of the algebra when exhibited in matrix form. For example, the seaweed algebra 2|4 1|2|3 consists of traceless matrices of the form depicted on the left side of Figure 2, where * 's indicate potential non-zero entries. Figure 2: A seaweed of type 2|4 1|2|3 and its associated meander The index of a Lie algebra was introduced by Dixmier [16]. Formally, the index of a Lie algebra g is defined by where B f is the associated skew-symmetric Kirillov form defined by B f (x, y) = f ([x, y]), for all x, y ∈ g. The index is an important algebraic invariant of the Lie algebrathough notoriously difficult to compute. However, in [15], Dergachev and A. Kirillov developed a combinatorial algorithm to compute the index of a seaweed subalgebra of sl(n) by counting the number of connected components of a certain planar graph, called a meander, associated to the seaweed.
To construct a meander, let a 1 |···|am b 1 |···|bt be a seaweed. Now label the n vertices of our meander as v 1 , v 2 , . . . , v n from left to right along a horizontal line. We then place edges above the horizontal line, called top edges, according to a 1 + · · · + a m as follows: Partition the set of vertices into a set partition by grouping together the first a 1 vertices, then the next a 2 vertices, and so on, lastly grouping together the final a m vertices. We call each set within a set partition a block. For each block in the set partition determined by a 1 + · · · + a m , add an edge from the first vertex of the block to the last vertex of the block, then add an edge between the second vertex of the block and the second to last vertex of the block, and so on within each block. More explicitly, given vertices v j , v k in a block of size a i , there is an edge between them if and only if j + k = 2(a 1 + a 2 + · · · + a i−1 ) + a i + 1.
In the same way, place bottom edges below the horizontal line of vertices according to the blocks in the partition determined by b 1 + · · · + b t (see the right side of Figure 2).
Every meander consists of a disjoint union of cycles and paths. The main result of [15] is that the index of a seaweed can be computed by counting the number and type of these components in its associated meander.
Theorem 4. (Dergachev and A. Kirillov, [15]) If p is a seaweed subalgebra of sl(n), then where C is the number of cycles and P is the number of paths in the associated meander.
Example 5. In the example of Figure 2, the meander associated to the seaweed 2|4 1|2|3 has no cycles and consists of a single path -so, has index zero, hence is Frobenius.
While Theorem 4 is an elegant combinatorial result it is difficult to apply in practice. However, Coll et al in [13] show that any meander can be contracted or "wound-down" to the empty meander through a sequence of graph-theoretic moves, each of which is uniquely determined by the structure of the meander at the time of move application. There are five such moves, only one of which affects the component structure of the meander graph and is therefore the only move capable of modifying the index of the meander. Since we will need the explicit winding-down moves in the proof of Theorem 16 we review the winding-down process.
Lemma 6 (Winding-down). Given a meander M of type a 1 |a 2 | · · · |a m b 1 |b 2 | · · · |b t , create a meander M by exactly one of the following moves.
For all moves, except the Component Elimination move, M and M have the same index.
the electronic journal of combinatorics 27(3) (2020), #P3.1 Example 7. In this example, the seaweed 17|3 10|4|6 is wound-down to the empty meander using the moves detailed in Lemma 6. In what follows, it is helpful to add a sixth index preserving transformation, F h , called a horizontal flip which takes M to a m | · · · |a 2 |a 1 b t | · · · |b 2 |b 1 .

(λ, 1 w(λ) )
In this section we investigate, for fixed i and varying n, the sequence of values defined by the number of partitions λ ∈ P(n) such that ind 1 w(λ) (λ) = n − i. We find that for each i, if c i n = |{λ ∈ P(n) : ind 1 w(λ) (λ) = n − i}|, then {c i n } ∞ n=i is eventually constant, stabilizing at c i , which turns out to be a well-known classical value (see Theorem 11). The following Table 1 illustrates c i n for small values of n and i.
By coloring partitions, we can better understand the c i 's. We will use two colors (red and blue), to color the parts of a given partition. When enumerating colored partitions, we will assume that two partitions which are identical, save for their coloring, will be considered different partitions. So, for example, the partition of the integer 2 given by (1, 1) is different from the partition of the integer 2 given by (1, 1). We also tacitly assume that in a given colored partition all blue parts of a given size precede all red parts of the same size (see Example 9). Such colored partitions are called two-colored partitions. Remark 10. Two-colored partitions make an appearance in Guptas's 1958 article [21] and have recently been connected to other diverse objects, such as quandles [8,9].
The generating function for the number of two-colored partitions of n is well-known and is given by The following theorem connects the current exposition to classical partition theory.  Proof. The case i = 1 is clear since, by Theorem 4, the only λ ∈ P(n) with ind 1 w(λ) (λ) = n − 1 is λ = 1 n . We show that for i > 1 and n 3i − 3, there is a bijective correspondence between M(i, n) = {λ ∈ P(n) : ind 1 w(λ) (λ) = n − i} and P 2 (i − 1) = {two-colored partitions of i − 1}. We do this in two steps.
Remark 12. In [24], Seo and Yee provide an alternative proof of Theorem 11 which relies on generating functions.

(λ, w(λ)) -the maximal parabolic case
As above, let λ = (λ 1 , . . . , λ m ) be an element of P(n). In this section, we consider the seaweed defined by the pair of compositions (λ, w(λ)). In contrast to the previous section, here we investigate the number of partitions λ such that ind w(λ) (λ) = 0. We naturally call such partitions, Frobenius partitions. The main theorem of this section, Theorem 16 and Remark 17, remarkably establish that if λ i 7, for i = 1, . . . , m, then the number of Frobenius partitions is eventually a periodic function of n.
We begin with three Lemmas which will be helpful in the proof of Theorem 16.
Lemma 13. Let g = a 1 |···|am m i=1 a i be a seaweed algebra. If there exists i < j − 1 such that i l=1 a l = m l=j a l , then g is not Frobenius. Proof. Applying the winding moves (F v ) followed by i applications of (P ) to the meander corresponding to g results in the meander corresponding to the seaweed algebra of type b 1 |a i |···|a 1 a i+1 |···|a j |···|am where b 1 = j−1 l=i+1 a l > 0; but this meander consists of at least two components, one corresponding to b 1 a i+1 |···|a j−1 , and the other a i |···|a 1 a j |···|am . Thus, by Theorem 4, ind(g) > 0.
Lemma 14. Let g = a 1 |···|am m i=1 a i be a seaweed algebra. If there exists more than two odd a i 's, then g is not Frobenius.
Proof. Each odd a i contributes a vertex of degree 1 to the meander corresponding to g.
Furthermore, each open path consists of exactly two vertices of degree 1, and no closed path contains a vertex of degree 1. So, if there are more than two odd a i 's, then the corresponding meander must contain more than one open path. Therefore, by Theorem 4, ind(g) > 0.
Proof. The proof heuristic is described as follows. We consider the possible partitions for each d 4 -except for those cases considered in Lemma 13 and Lemma 14 -and determine which partitions are Frobenius. to the corresponding meander results in the meander for a partition of the form (1 1 4 f 4 ), which was found to be Frobenius in case 5 above.
Thus, there are two Frobenius partitions with weight congruent to 1(mod 4); two with weight congruent to 2(mod 4); and one with weight congruent to 3(mod 4).

Conclusion
We use the recent index theory of seaweed algebras to generate partition statistics. Starting with a partition λ, we form a seaweed algebra and define the index of the latter to be the index of the former. Such seaweeds (and attendant index-based statistics) are defined by pairing λ with a λ-based composition. We consider the two extremal cases defined by (λ, 1 w(λ) ) and (λ, w(λ)).
• In studying pairs (λ, 1 w(λ) ) we establish a relation between our new index based statistics and the more classical theory of integer partitions in the form of twocolored partitions.
• As noted in the Introduction, the enumeration of Frobenius seaweeds is of keen topical interest. In studying pairs (λ, w(λ)), we find, remarkably, that the number of such Frobenius pairs is eventually periodic.
This investigation raises more questions and directions for study. We list a few.

(Congruence relations)
Historically, statistics on partitions were motivated by their utility to witness congruence relations [3,5,6,18]. Do index-based statistics have similar classical value?

(Weighted sums) After the fashion of Euler's Pentagonal Number Theorem [2]
and other Legendre type theorems [4], index-based statistics could also be incorporated into weighted sums. Normally in Legendre type theorems, partitions λ contribute a term of the form (−1) l(λ) q w(λ) to the weighted sum, where l(λ) is the number of parts of λ. One could instead insist that each partition contributes a term of the form (−1) ind w(λ) (λ) q w(λ) . For example, by restricting to partitions with only odd parts (denoted P(n, S odd )), considering the sets e n = |{λ ∈ P(n, S odd ) : ind w(λ) (λ) is even}| and o n = |{λ ∈ P(n, S odd ) : ind w(λ) (λ) is odd}|, and insisting that all terms in the weighted sum are non-negative, numerical data suggests the following interesting conjecture.