Refined Cyclic Sieving on Words for the Major Index Statistic
Connor Ahlbach, Joshua Swanson

TL;DR
This paper introduces a combinatorial refinement of the cyclic sieving phenomenon related to the major index statistic on words with fixed content and cyclic descent type, providing a new bijective proof approach.
Contribution
It formulates and proves a combinatorial refinement of CSP for the major index statistic, differing from previous representation-theoretic methods, and introduces the universal sieving statistic "flex".
Findings
Refined CSP for the major index on words with fixed content.
A new combinatorial, bijective proof approach.
Extension of cyclic sieving to shifted subset sums.
Abstract
Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action and a polynomial. A key example arises from the length generating function for minimal length coset representatives of a parabolic quotient of a finite Coxeter group. In type A, this result can be phrased in terms of the natural cyclic action on words of fixed content. There is a natural notion of refinement for many CSP's. We formulate and prove a refinement, with respect to the major index statistic, of this CSP on words of fixed content by also fixing the cyclic descent type. The argument presented is completely different from Reiner-Stanton-White's representation-theoretic approach. It is combinatorial and largely, though not entirely, bijective in a sense we make precise with a "universal" sieving statistic on words, "flex". A building block of our argument involves…
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Refined Cyclic Sieving on Words
for the Major Index Statistic
Connor Ahlbach and Joshua P. Swanson
Department of Mathematics, University of Washington, Seattle, WA 98195, USA
Abstract.
Reiner–Stanton–White [RSW04] defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action and a polynomial. A key example arises from the length generating function for minimal length coset representatives of a parabolic quotient of a finite Coxeter group. In type , this result can be phrased in terms of the natural cyclic action on words of fixed content.
There is a natural notion of refinement for many CSP’s. We formulate and prove a refinement, with respect to the major index statistic, of this CSP on words of fixed content by also fixing the cyclic descent type. The argument presented is completely different from Reiner–Stanton–White’s representation-theoretic approach. It is combinatorial and largely, though not entirely, bijective in a sense we make precise with a “universal” sieving statistic on words, .
A building block of our argument involves cyclic sieving for shifted subset sums, which also appeared in Reiner–Stanton–White. We give an alternate, largely bijective proof of a refinement of this result by extending some ideas of Wagon–Wilf [WW94].
Accepted to the European Journal of Combinatorics. ©2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/.
1. Introduction
Since Reiner, Stanton, and White introduced the cyclic sieving phenomenon (CSP) in 2004 [RSW04], it has become an important companion to any cyclic action on a finite set. Some remarkable examples of the CSP involve the action of a Springer regular element on Coxeter groups [RSW04, Theorem 1.6], the action of Schutzenberger’s promotion on Young tableaux of fixed rectangular shape [Rho10], and the creation of new CSPs from old using multisets and plethysms with homogeneous symmetric functions [BER11, Proposition 8]. See [Sag11] for Sagan’s thorough introduction to the cyclic sieving phenomenon. More recent work on the CSP includes [ARR15, Pec14, PSV16]. Here we are concerned with cyclic sieving phenomena involving cyclic descents on words. Cyclic descents were used implicitly by Klyachko [Kly74] and independently introduced by Cellini [Cel98]. Since then, cyclic descents have been used by Lam and Postnikov in studying alcoved polytopes [LP12] and by Petersen in studying -partitions [Pet05]. They also appear prominently in an ongoing line of research on cyclic descent extensions for standard tableaux by Adin, Elizalde, Reiner, and Roichman [ER17, ARR17, AER17].
An earlier “extended abstract” for the present work appeared in [AS17]. We assume some familiarity with the CSP, though we recall certain key statements.
Definition 1.1**.**
Suppose is a cyclic group of order generated by , is a finite set on which acts, and . We say the triple exhibits the cyclic sieving phenomenon (CSP) if for all ,
[TABLE]
where is any fixed primitive -th root of unity.
Representation theoretically, evaluations of at -th roots of unity yield the characters of the -action on .
In many instances of cyclic sieving, and all of those considered here, is the generating function for some statistic on . Given a statistic , let
[TABLE]
We say two statistics are equidistributed on if , and we say they are equidistributed modulo on if .
Our main result is a refinement of a CSP triple first observed by Reiner–Stanton–White, which we now summarize; see Section 2 for missing definitions. Consider words in the alphabet . Given a word of length , let denote the content of and write
[TABLE]
for the set of words with content . Write for the major index of . The cyclic group acts on words of length by rotation. The following expresses an interesting result of Reiner, Stanton, and White in our notation.
Theorem 1.2**.**
[RSW04*, Proposition 4.4]**.
Let . The triple*
[TABLE]
exhibits the CSP.
Reiner, Stanton, and White deduced Theorem 1.2 from the following more general result about Coxeter systems.
Theorem 1.3**.**
[RSW04*, Theorem 1.6]**.
Let be a finite Coxeter system and . Let be the corresponding parabolic subgroup, the set of minimal length representatives for left cosets , and . Let be a cyclic subgroup of generated by a Springer regular element. Then exhibits the cyclic sieving phenomenon.*
Theorem 1.2 follows from Theorem 1.3 when by identifying with words of fixed content , where is the composition recording the lengths of consecutive subsequences of , and is generated by an -cycle. One must also use the classical result of MacMahon that is equidistributed with the inversion statistic on words, from which it follows that [Mac13, Art. 6].
Definition 1.4**.**
A refinement of a CSP triple is a CSP triple
[TABLE]
where has the restricted -action.
If refines , then so does where . Thus, a CSP refinement partitions into smaller CSPs with the same statistic. If is an orbit, its only refinements are and . In Section 8, we define a statistic on words, flex, which is universal in the sense that it refines to all -orbits. Such universal statistics are essentially equivalent to the choice of a total ordering for each orbit of .
We partition words of fixed content into fixed cyclic descent type (). One computes by building up by adding all ’s, ’s, , and counting the number of cyclic descents introduced at each step. For precise details, see Definition 4.1 and Example 4.2. We write the set of words with fixed content and cyclic descent type as
[TABLE]
Our main result is the following.
Theorem 1.5**.**
Let and be any composition. The triple
[TABLE]
refines the CSP triple .
It is not at all clear how to modify Reiner–Stanton–White’s representation-theoretic approach to Theorem 1.2 to give Theorem 1.5, since is not closed under the -action. Finding a representation-theoretic interpretation of Theorem 1.5 would be quite interesting.
In the course of proving Theorem 1.5, we derive an explicit product formula for mod involving -binomial coefficients, Theorem 5.19. The formula results in a -identity similar to the Vandermonde convolution identity; see Corollary 5.20. The argument involves constructing algorithmically by recursively building a certain tree.
The two-letter case of Theorem 1.5 can be rephrased as follows. Fix and . Let denote the set of subsets of of size where . Define the statistic by identifying with and setting , which sums the maximum of the cyclic blocks of .
Corollary 1.6**.**
The triple
[TABLE]
exhibits the CSP.
Example 1.7**.**
When ,
[TABLE]
which have statistic , respectively, so . We then have , in agreement with (1).
Theorem 8.3 in [RSW04] and hence Theorem 1.2 builds on a representation-theoretic result due to Springer [Spr74, Proposition 4.5]. Our argument is highly combinatorial, but it is not entirely bijective. Finding an explicit bijection would be quite interesting. See Section 8 for more details.
A key building block of our proof of Theorem 1.5 involves cyclic sieving on multisubsets and subsets, which was also first stated in [RSW04]. We describe refinements of these results as well, Theorem 7.4 and Theorem 7.11, restricting to certain gcd requirements in the subset case. We present a completely different inductive proof of our subset refinement in the spirit of our proof of Theorem 1.5. Both our proof of Theorem 7.11 and Theorem 1.5 use an extension lemma, Lemma 3.3, which allows us to extend CSPs from smaller cyclic groups to larger ones.
The rest of the paper is organized as follows. In Section 2, we recall combinatorial background. In Section 3, we introduce the concept of modular periodicity and prove our extension lemma, Lemma 3.3. In Section 4, we define cyclic descent type. In Section 5, we decompose words with fixed content and cyclic descent type and prove a product formula for modulo , Theorem 5.19. Section 6 uses the results of Section 5 to prove our main result, Theorem 1.5. Section 7 refines cyclic sieving on multisubsets and subsets with respect to shifted sum statistics. In Section 8, we introduce the flex statistic and use it to reinterpret Theorem 1.5.
2. Combinatorial Background
In this section, we briefly recall or introduce combinatorial notions on words and fix our notation. We use the alphabet of positive integers throughout unless otherwise noted. We also write or for the cardinality of a set . For , recall the notation
[TABLE]
A word of length is a sequence of letters . Let denote the length of a word . Let denote the set of all words of length . The descent set of is , and the number of descents is . The major index of is . The cyclic descent set of is , where now the subscripts are taken mod , and we write for the number of cyclic descents. Any position that is not a cyclic descent is a cyclic weak ascent. The inversion number of is . We use lower dots between letters to indicate cyclic descents and upper dots to indicate cyclic weak ascents throughout the paper as in the following example.
Example 2.1**.**
If , then , , , , , , and .
A composition or weak composition of is a sequence of non-negative integers summing to , typically denoted . A composition is strong if for all . The content of a word , denoted , is the sequence whose -th part is the number of ’s in . For , is a weak composition of . We write
[TABLE]
The cyclic group of order acts on by rotation as
[TABLE]
Typically we consider to be the long cycle .
The set of all words in is a monoid under concatenation. A word is primitive if it is not a power of a smaller word. Any non-empty word may be written uniquely as for with primitive. We call the period of , written , and the frequency of , written . An orbit of under rotation is a necklace, usually denoted . We have and . Content, primitivity, period, frequency, and are all constant on necklaces.
Example 2.2**.**
The necklace of is
[TABLE]
which has period , frequency , and .
Reiner–Stanton–White gave equivalent conditions for a triple to exhibit the CSP. In place of (1) in Definition 1.1, we may instead require
[TABLE]
where the sum is over all orbits under the action of on . Note that for ,
[TABLE]
This means every -action on a finite set gives rise to a CSP , where is the right hand side of (5). We refer the interested reader to [RSW04, Proposition 2.1] for the proof of the equivalence of (1) and (5).
Remark 2.3**.**
If exhibits the CSP, then so do both of the triples and when by (1). In the latter case we have relaxed the constraint to , which does no harm since (1) involves evaluations at roots of unity. Further, if and exhibit the CSP, then and exhibit the CSP, where acts on by [BER11, Prop. 2.2].
For a set , write
[TABLE]
Let . We use the following standard -analogues:
[TABLE]
We write . Observe that the cyclic group of order acts on by . This induces actions of on and \left.\mathchoice{\left(\kern-4.79996pt\binom{[0,n-1]}{k}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[0,n-1]}}{\smash{k}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[0,n-1]}}{\smash{k}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[0,n-1]}}{\smash{k}}\kern-3.00003pt\right)}\right. by acting on values in each subset or multisubset. For example, . These actions, in slightly more generality, appear in one of the original, foundational CSP results as follows.
Theorem 2.4**.**
[RSW04*, Thm. 1.1]**.
In the notation above, the triples*
[TABLE]
exhibit the CSP.
We will also have use of the following principal specializations (see [Mac95, Example I.2.2] or [Sta99, Proposition 7.8.3]):
[TABLE]
Here the statistic denotes the sum of the elements of a subset or submultiset of .
Recall that the length function on coincides with the inversion statistic defined above on words of content . More generally, minimal length coset representatives of parabolic quotients also have length given by the inversion statistic on the corresponding words . The following classical result is due to MacMahon.
Theorem 2.5**.**
[Mac13*, Art. 6]**.
For each , and are equidistributed on with*
[TABLE]
3. Modular Periodicity and an Extension Lemma
We now introduce the concept of modular periodicity and use it to give an extension lemma, Lemma 3.3, which allows us to extend CSP’s from certain subgroups to larger groups. We will verify the hypotheses of Lemma 3.3 in the subsequent sections to deduce Theorem 1.5.
Definition 3.1**.**
We say a statistic has period modulo on if for all ,
[TABLE]
Similarly, we say a Laurent polynomial has period modulo if
[TABLE]
or equivalently if .
For example, has period 2 modulo . Note that has period modulo on if and only if has period modulo . The following basic properties of modular periodicity will be useful throughout the paper.
Lemma 3.2**.**
Let and .
- (i)
If has period modulo and period modulo , then has period modulo for any . In particular, has period modulo . 2. (ii)
If has period modulo and period modulo , then has period modulo . 3. (iii)
If has period modulo and , then has period modulo . 4. (iv)
If has period modulo , then so does for any Laurent polynomial . 5. (v)
If has period modulo and , then
[TABLE]
Proof.
(i), (iii), (iv), and (v) are straightforward. For (ii), suppose
[TABLE]
Write . If does not divide , then it must divide and hence . It follows that
[TABLE]
Lemma 3.3**.**
Suppose acts on . Let and . If
- (i)
* exhibits the CSP,* 2. (ii)
* has period modulo , and* 3. (iii)
for all -orbits , we have ,
then exhibits the CSP.
Proof.
Let
[TABLE]
By (5), exhibits the CSP, so also exhibits the CSP by Remark 2.3. Thus, by (5) and condition (i),
[TABLE]
for some . Each summand of has period modulo since
[TABLE]
by condition (iii). Putting this together with condition (ii), and have period modulo . Using Lemma 3.2(v) twice along with (11) now gives
[TABLE]
4. Cyclic Descent Type
In this section, we introduce the cyclic descent type of a word. We also verify hypothesis (iii) of Lemma 3.3 for for a particular ; see Lemma 4.4.
Let denote the subsequence of with all letters larger than removed. We have a “filtration”
[TABLE]
where means that is a subsequence of . We think of this filtration as building up by recursively adding all of the copies of the next largest letter “where they fit.” The cyclic descent type of a word , denoted , is the sequence which tracks the number of new cyclic descents at each stage of the filtration. Precisely, we have the following.
Definition 4.1**.**
The cyclic descent type (CDT) of a word is the weak composition of given by
[TABLE]
Note that is constant on necklaces since rotating rotates each and is constant under rotations. Furthermore, always, so always begins with [math].
Example 4.2**.**
Suppose , so
[TABLE]
Hence, .
Recall from (4) that
[TABLE]
We could define more “symmetrically” by replacing with “cyclic weak ascent type,” which would be the point-wise difference of and . However, content is ubiquitous in the literature, so we use it.
Remark 4.3**.**
Despite (10), and are not equidistributed even modulo on in general, so does not generally exhibit the CSP. For example, , which has
[TABLE]
which are not even congruent modulo .
Lemma 4.4**.**
If , , is a necklace, and , then .
Proof.
Suppose is the necklace of , meaning , so we can write . Hence, using pointwise multiplication,
[TABLE]
In particular, divides , so .
5. Runs and Falls
In this section, we give a method to algorithmically construct and use it to prove a product formula for modulo , Theorem 5.19. We conclude the section by using this formula to verify hypothesis (ii) of Lemma 3.3 for ; see Proposition 5.21.
5.1. A Tree Decomposition for
We now describe a way to create words with a fixed content and CDT in terms of insertions into runs and falls. This procedure is organized into a tree, Definition 5.11, whose edges are labeled with sets and multisets. Lemma 5.8 describes changes in the major index upon traversing an edge of this tree.
Definition 5.1**.**
Write . A fall in is a maximal set of distinct consecutive indices such that , where we take indices modulo . A run in a non-constant word is a maximal set of distinct consecutive indices such that , where we take indices modulo . The constant word by convention has no runs and falls.
Note that each letter in is part of a unique fall and a unique run, except when is constant. It is easy to see that has falls and runs, since they are separated by cyclic weak ascents and cyclic descents, respectively. Note that this holds if is constant since then by convention has no runs. Index falls from [math] from left to right starting at the fall containing the first letter of , and do the same with runs.
Definition 5.2**.**
We write
[TABLE]
for the indices of the falls and runs of , respectively.
Example 5.3**.**
Let , where upper dots indicate cyclic weak ascents and lower dots indicate cyclic descents. Since , we have and . The falls of are , with respective indices . The runs of are , with respective indices .
Definition 5.4**.**
Let be a word. Fix a letter and pick a subset of the falls . Assume does not appear in any of the falls in . We insert into falls by successively inserting into each fall in so that is still decreasing.
Similarly, we may fix a letter and pick a multisubset of (this time may already appear in a run in ). We insert into runs by successively inserting into each run in so that is still weakly increasing.
When inserting into a run already containing , the resulting word is independent of precisely which of the possible positions is used. This is the reason we insert into runs and falls instead of positions.
Note that there is a slight ambiguity in our description of insertion into falls and runs, since it may be possible to insert either at the beginning or at the end of while still satisfying the relevant inequalities. Given the choice, we always insert at the beginning of .
Example 5.5**.**
Let . Insert into falls of with indices [math] and to successively obtain and then . Note that has two more runs (or cyclic descents) than . Now insert into the runs of with multiset of indices to successively obtain , , , and .
Let
[TABLE]
We restrict to and since the major index generating function is easier to find and extends to (mod ).
Definition 5.6**.**
Fix , a letter not in , and
[TABLE]
where denotes a multisubset. Let be obtained by inserting into falls of . Note that indexes the runs of . Now let be obtained by inserting into runs of . We say is obtained by inserting the triple into . Observe that and .
We next describe the effect of inserting a single letter on . We restrict to so we preserve a cyclic weak ascent at the end and never add a letter to the end. The fact that the increments in major index from inserting a new letter into all possible positions form a permutation was first observed by Gupta [Gup78]. Lemma 5.7 tells us exactly the increment in major index based on which run or fall the newly inserted letter fits into.
Lemma 5.7**.**
Suppose is obtained by adding a letter to in any position. Then is obtained by inserting into some run or fall of , and
[TABLE]
Proof.
If , then is inserted into some run of , and otherwise and is inserted into some fall of . Inserting into run of will increment the position of descents by each, so
[TABLE]
Let , which is the sum of where . Inserting into fall of will increment the position of weak ascents by each, so
[TABLE]
from which it follows that
[TABLE]
Lemma 5.8**.**
Suppose is obtained by inserting the triple into . Then
[TABLE]
Proof.
Let be obtained by inserting into falls of . It suffices to show
[TABLE]
and
[TABLE]
since . Both (17) and (18) follow from iterating Lemma 5.7 and recalling is incremented by 1 each time we insert into a fall.
Notation 5.9**.**
For the rest of this section, fix a strong composition of and with . We emphasize that and have the same number, , of parts. For , let
[TABLE]
For , we have the defining conditions and . Furthermore, let
[TABLE]
and
[TABLE]
If , then the set consists of all subsets of the falls which, when is inserted into those falls of , result in a word with cyclic descents. The multiset similarly consists of all choices of runs which, when is inserted into those runs, result in a word with length .
Remark 5.10**.**
We restrict to strong compositions for notational simplicity, though the results in this section may easily be generalized to arbitrary weak compositions by “flattening” weak compositions to strong ones by removing zeros.
Definition 5.11**.**
Construct a rooted, vertex-labeled and edge-labeled tree recursively as follows. Begin with a tree containing only a root labeled by the word . For , to obtain , do the following. For each leaf of and for each triple with
[TABLE]
add an edge labeled by to from to where is obtained by inserting into . Define .
Example 5.12**.**
Let and . Figure 1 is the tree .
Let and . Figure 2 is the subgraph of consisting of paths from the root to leaves that are rotations of . For this full , the root has children since has falls. Each child of the root itself has \binom{4}{1}\left.\mathchoice{\left(\kern-4.79996pt\binom{3}{2}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{3}}{\smash{2}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{3}}{\smash{2}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{3}}{\smash{2}}\kern-3.00003pt\right)}\right.=24 children. Hence, has leaves. Notice that the cyclic rotations of appearing as leaves in Figure 2 are precisely those ending in . It will shortly become apparent that in this example, .
Lemma 5.13**.**
The vertices of which are edges away from the root are precisely the elements of , each occurring once. In particular, the leaves of are precisely the elements of , each occurring once.
Proof.
By definition of and , any leaf of has content , cyclic descent type , and ends in a 1, so is in . Conversely, given any , the word is obtained by inserting a unique triple into by repeated applications of Lemma 5.7.
Definition 5.14**.**
By Lemma 5.13, the tree encodes a bijection
[TABLE]
given by reading the edge labels from the root to . We suppress the dependence of on and from the notation since they can be computed from the input .
Lemma 5.15**.**
For any ,
[TABLE]
Consequently,
[TABLE]
Proof.
Each has distinct cyclic rotations, of which end in .
Proposition 5.16**.**
Using Notation 5.9, we have
[TABLE]
In particular, if and only if
[TABLE]
Proof.
The product in (23) is , which is by the bijection . Now (23) follows from (22), and (24) follows from (23).
5.2. Major Index Generating Functions
We next use the bijection and Lemma 5.8 to give a product formula for , Theorem 5.17. We then use modular periodicity to obtain an analogous expression for modulo , Theorem 5.19.
Theorem 5.17**.**
Using Notation 5.9, we have
[TABLE]
where
[TABLE]
Proof.
Recall denotes the sum of the elements of a set or multiset . Combining with Lemma 5.8 shows that
[TABLE]
where
[TABLE]
Noting that
[TABLE]
simplifying (27) gives
[TABLE]
Equation (25) now follows from (8), (9), and the definition of and . As for (26), consider the reversal bijection induced by
[TABLE]
on . This bijection satisfies , so
[TABLE]
Plugging (29) into (28) and noting that
[TABLE]
gives
[TABLE]
Using (8) and (9) now yields (26).
Lemma 5.18**.**
Let , . The statistic has period modulo on . Moreover, is constant modulo on necklaces in , and
[TABLE]
Proof.
Since cyclically rotating increments each cyclic descent by modulo , we have
[TABLE]
In particular, has period modulo on necklaces in . Furthermore, is constant on necklaces in modulo . By (21), each necklace has the same fraction, , of its elements in , so (30) follows.
Theorem 5.19**.**
Using Notation 5.9, let . Then, modulo ,
[TABLE]
Proof.
By Lemma 5.18, has period modulo on . Hence by Lemma 3.2(i), has period modulo on . Using Lemma 3.2(v) and (30) gives
[TABLE]
where . Theorem 5.19 now follows from Theorem 5.17.
Corollary 5.20**.**
Using Notation 5.9, let . Then, modulo ,
[TABLE]
where the sum is over weak compositions of satisfying (24). In particular,
[TABLE]
Note that the two-letter case of (34) is a special case of the classical Vandermonde convolution identity [Sta12, Ex. 1.1.17].
5.3. Verifying Hypothesis (ii) of Lemma 3.3 for
Proposition 5.21**.**
Using Notation 5.9, has period modulo .
Proof.
Let . By Theorem 5.19,
[TABLE]
modulo . The action of rotation on elements of increases their sum by modulo . Thus by (8), has period modulo . Similarly by (9), \left.\mathchoice{\left(\kern-4.79996pt\binom{k_{\ell}}{\alpha_{\ell}-\delta_{\ell}}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{k_{\ell}}}{\smash{\alpha_{\ell}-\delta_{\ell}}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{k_{\ell}}}{\smash{\alpha_{\ell}-\delta_{\ell}}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{k_{\ell}}}{\smash{\alpha_{\ell}-\delta_{\ell}}}\kern-3.00003pt\right)}\right._{q^{-1}} has period modulo . For , by Lemma 3.2(iv) we then have
[TABLE]
We show has period and modulo by downward induction on , for . Note that the base case is accounted for by our argument as well.
Suppose has period and modulo for all . By Lemma 5.18, has period modulo . By Lemma 3.2(i), thus has period
[TABLE]
modulo . Since has period modulo , has period modulo by Lemma 3.2(ii).
As noted, has period modulo . By Lemma 3.2(i), also has period
[TABLE]
modulo . Hence, as has period modulo , has period modulo by Lemma 3.2(ii). By another application of Lemma 3.2(i), has period modulo as well, completing the induction.
Indeed, has period modulo trivially, and has period modulo by Lemma 3.2(i). Putting everything together, has periods modulo , so by one more application of Lemma 3.2(i), has period modulo .
6. Refining the CSP to fixed content and Cyclic Descent Type
In this section, we verify the final hypothesis (i) of Lemma 3.3 for and deduce Theorem 1.5. Throughout this section we continue to follow Notation 5.9. We recall in particular that
[TABLE]
and
[TABLE]
6.1. A Fixed Point Lemma
To prove our main result, Theorem 1.5, one approach would be to find a -equivariant isomorphism between a known CSP triple and . Such a triple is hinted at by (25) and the bijection using products of CSP’s coming from Theorem 2.4, though the approach encounters immediate difficulties. For instance, is not generally closed under the -action. In this section, we instead give a fixed point lemma, Lemma 6.5, which is intuitively a weakened version of the equivariant isomorphism approach.
Definition 6.1**.**
We define -actions on , , and as follows. Since , , and , acts on each of , , and by restricting the actions of , , and to their unique subgroups of size . For instance, the action of on is generated by rotation by .
We additionally define -actions on and by letting act diagonally. We emphasize that despite having -actions on and , the bijection is not in general equivariant since is not closed under the action on .
Definition 6.2**.**
Given a multisubset of some set , we may encode it as a multiplicity word where is the multiplicity of . In particular, we may consider the bijection as mapping words to sequences of pairs of certain words.
Example 6.3**.**
Consider the leaf in Figure 2 from Example 5.12. Reading edge labels gives . Recalling that consists of subsets of , consists of multisubsets of , consists of subsets of , and consists of multisubsets of , the corresponding sequence of words is , where denotes the empty word. Table 1 summarizes several similar translations.
Lemma 6.4**.**
Suppose for some word . If
[TABLE]
encoded as multiplicity words as in Definition 6.2, then
[TABLE]
Proof.
The insertion triples needed to build are the sequences of shifted copies of the insertion triples needed to build .
Lemma 6.5**.**
An element fixes if and only if fixes .
Proof.
For , let denote the order of . It is easy to see that fixes if and only if there is some word such that .
Suppose fixes , so that . By Lemma 6.4,
[TABLE]
Each of the words and is fixed by , so is fixed by . The reverse implication follows analogously using the fact that is a bijection.
6.2. Verifying Hypothesis (i) of Lemma 3.3 for
Theorem 6.6**.**
Using Notation 5.9, exhibits the CSP.
Proof.
We use the notation and actions in Definition 6.1. Recall that
[TABLE]
From Theorem 2.4, for each ,
[TABLE]
exhibit the CSP. Taking products,
[TABLE]
exhibits the CSP. Comparing this to Theorem 5.17, we have
[TABLE]
modulo , as because for all . Thus,
[TABLE]
exhibits the CSP.
By Lemma 5.15, for any ,
[TABLE]
Since is an orbit under , an element fixes if and only if fixes pointwise. Thus, for any ,
[TABLE]
Combining (38) and Lemma 6.5 now shows that for any ,
[TABLE]
Hence, by (39), the CSP in (37), and (1),
[TABLE]
exhibits the CSP. By (30), modulo , hence also modulo since , completing the proof.
We have now finished the verification of the conditions in Lemma 3.3 for . Condition (i) is Theorem 6.6, Condition (ii) is Proposition 5.21, and Condition (iii) is Lemma 4.4. This completes the proof of Theorem 1.5.
7. Refinements of Binomial CSP’s
A key step in the proof of Theorem 6.6 was Theorem 2.4 due to Reiner–Stanton–White, which says that the triples
[TABLE]
exhibit the CSP. Indeed, [RSW04] contains two proofs, one via representation theory [RSW04, §3] and another by direct calculation [RSW04, §4]. In this section, we give two refinements of related CSP’s involving an action of on sets of subsets (Theorem 7.11) and multisubsets (Theorem 7.4) for all , using shifted sum statistics. Our proof of the subset refinement, Theorem 7.11, does not use Theorem 2.4, so it can be used as an alternative proof of the subset case of Theorem 2.4. Our method is inspired by the rotation of subintervals used by Wagon and Wilf in [WW94, §3].
7.1. Cyclic Actions and Notation
We define two different cyclic actions of the cyclic group of order on and induce these actions to and \left.\mathchoice{\left(\kern-4.79996pt\binom{[0,n-1]}{k}\kern-4.79996pt\right)}{\big{(}\kern-3.00003pt\binom{\smash{[0,n-1]}}{\smash{k}}\kern-3.00003pt\big{)}}{\left(\kern-3.00003pt\binom{\smash{[0,n-1]}}{\smash{k}}\kern-3.00003pt\right)}{\left(\kern-3.00003pt\binom{\smash{[0,n-1]}}{\smash{k}}\kern-3.00003pt\right)}\right.. We also fix notation for the rest of the section.
Notation 7.1**.**
Fix , and . Let
[TABLE]
For all , let
[TABLE]
which we call a -interval. For any composition with parts, let
[TABLE]
where the intersection in (41) preserves the multiplicity of . We also fix cyclic groups , of order whose actions are described below.
Let act on by simultaneous rotation of -intervals, which is generated by the permutation
[TABLE]
in cycle notation. On the other hand, has a unique subgroup of order which also acts on and is generated by the permutation
[TABLE]
Induce these actions of and up to and by
[TABLE]
Notice that the action of restricts to and for any .
Let be a pair where is a group acting on a set . A morphism of group actions is a pair where is a group homomorphism and is a map of sets which satisfy
[TABLE]
Remark 7.2**.**
The actions of and on are isomorphic since and have the same cycle type. This isomorphism explicitly arises from with , , etc. Thus the actions of and on and are isomorphic as well.
Recall the statistic sums the elements of a set or multiset. We also use the following shifted sum statistic. For , let
[TABLE]
[TABLE]
Using (45), we may restate Theorem 2.4 as saying that
[TABLE]
exhibit the CSP. Moreover, under the restricted action of on and ,
[TABLE]
exhibit the CSP by Remark 2.3. By Remark 7.2,
[TABLE]
also exhibit the CSP.
Example 7.3**.**
Let , , and . Abbreviating as , etc., gives
[TABLE]
Here, acts on by the permutation , and acts by . contains in addition to, for instance, .
7.2. A Multisubset Refinement
We next prove a refinement of the CSP triple in (46) by fixing sizes of intersections with the -intervals.
Theorem 7.4**.**
Recall Notation 7.1, and fix a composition . Then, refines the CSP triple .
Proof.
Separating the -intervals into different multisubsets gives
[TABLE]
which preserves the natural -action and statistic modulo . Since
[TABLE]
exhibits the CSP for all , the result follows from Remark 2.3.
The following analogous result holds for subsets.
Proposition 7.5**.**
Recall Notation 7.1, and additionally fix a composition . Then exhibits the CSP, where
[TABLE]
Proof.
Separating the -intervals into different subsets gives
[TABLE]
which preserves the -action and statistic modulo . Since
[TABLE]
exhibits the CSP for all , exhibits the CSP by Remark 2.3.
Remark 7.6**.**
Since we must shift the statistic by different amounts depending on , Proposition 7.5 is not a CSP refinement, in contrast to Theorem 7.4.
7.3. A Subset Refinement
We next prove an honest refinement of the CSP triple in (46). To do so, we restrict to certain subsets of for each divisibility chain ending in . Our proof again inductively extends CSP’s up from cyclic subgroups of using Lemma 3.3. In this subsection we first define our restricted subsets and give some examples. We then present a series of lemmata verifying the conditions of Lemma 3.3 before proving our refinement, Theorem 7.11.
Definition 7.7**.**
Suppose . Let
[TABLE]
We have and for all other .
Remark 7.8**.**
By conditioning on the sizes of the intersections with -intervals, decomposes as the disjoint union
[TABLE]
ranging over all satisfying
[TABLE]
Example 7.9**.**
If , then abbreviating as , etc., gives
[TABLE]
Consequently, and .
Definition 7.10**.**
Suppose is a totally ordered chain in the divisibility lattice ending with , i.e. where . Write
[TABLE]
We may now state our subset refinement. The proof is postponed to the end of this subsection.
Theorem 7.11**.**
Using Notation 7.1, let be a totally ordered chain in the divisibility lattice ending with and starting with . Then, refines the CSP triple .
Example 7.12**.**
If , , and , then has orbits and . Moreover,
[TABLE]
so exhibits the CSP by (5).
In fact, the subset case of Theorem 2.4 is the special case of Theorem 7.11, so the proof below of Theorem 7.11 yields an alternative proof of the subset case of Theorem 2.4.
Corollary 7.13**.**
* exhibits the CSP.*
Lemma 7.14**.**
Let be a totally ordered chain in the divisibility lattice ending with and beginning with . Suppose is the unique subgroup of of order .
- (i)
, where the disjoint union is over a subset of the sequences satisfying and . 2. (ii)
* is closed under the and -actions on .* 3. (iii)
The and -actions on are isomorphic. 4. (iv)
For any -orbit of , we have . 5. (v)
The statistic has period modulo on .
Proof.
For (i), by (51) we have where
[TABLE]
Write . For all , since , each -interval is a union of -intervals. Thus, for , whether for any depends only on , so or . Now (i) follows from .
For (ii), by (i) it suffices to show that each is closed under the and -actions. Since rotates -intervals, it preserves the size of each -interval, so indeed maps to itself. The same argument applies with in place of .
For (iii), by (i), it suffices to show the and -actions on are isomorphic. Recalling (49), we have
[TABLE]
By Remark 7.2, the actions of and on are isomorphic for each , so their actions on are isomorphic as well.
For (iv), pick with for as in (i). Let , which has elements. Viewing as a multiplicity word as in Definition 6.2, we see that has zeros and ones. For all , is some word repeated times. Using the two-letter case of Lemma 4.4, we have . Thus .
For (v), it suffices to show that has period modulo on for as in (i). By the gcd condition, there exist such that
[TABLE]
For some particular , consider cyclically rotating the elements of forward by in for all . The result is a bijection that satisfies , from which (v) follows.
Example 7.15**.**
Let , and . Then
[TABLE]
We have where and . Similarly where and . In fact,
[TABLE]
since, for instance, when while when .
Lemma 7.16**.**
Let . The action on is trivial and
[TABLE]
exhibits the CSP.
Proof.
All subsets in have each -interval either full or empty, so fixes every . By (5), thus exhibits the CSP if and only if mod . If the result is trivial, so take . For any , since each -interval is full or empty, we have and
[TABLE]
We may now prove Theorem 7.11.
Proof of Theorem 7.11.
We induct on . If , then the relevant triple exhibits the CSP trivially. For the induction step, we first claim that exhibits the CSP. If , then , so by Lemma 7.16 the action is trivial. It is easy to see that CSP’s with trivial actions refine to arbitrary subsets, so exhibits the CSP in this case. If , by conditioning on the sizes of the intersections of the -intervals, we can write
[TABLE]
where denotes the chain with prepended to . Hence exhibits the CSP by induction for each , since begins with . Thus exhibits the CSP by (53), proving the claim.
In order to realize the CSP triple from the CSP triple , we verify the conditions of Lemma 3.3. From Lemma 7.14(ii), the restriction of the -action on to the subgroup of size is isomorphic to the -action on , giving Condition (i). Condition (ii) is Lemma 7.14(v), and Condition (iii) is Lemma 7.14(iv). Thus exhibits the CSP by Lemma 3.3.
8. The Flex Statistic
We conclude by formalizing the notion of universal sieving statistics and giving an example, flex, in the context of words. We end with an open problem.
Definition 8.1**.**
Given a set with a -action, we say is a universal CSP statistic for if exhibits the CSP for all -orbits of .
Definition 8.2**.**
Let denote the index at which appears when lexicographically ordering the necklace , starting from [math]. Let flex be the product
[TABLE]
For example, listing in lexicographic order gives
[TABLE]
so, noting , we have
[TABLE]
Lemma 8.3**.**
The function is a universal CSP statistic for .
Proof.
Let be any necklace of length words. Since , and , we have
[TABLE]
so exhibits the CSP by (5).
Given a universal sieving statistic on some set , takes on precisely the values modulo on any orbit of size . The converse holds as well. In this sense, up to shifting values by , universal sieving statistics are equivalent to total orderings on each orbit of .
Standing in contrast to Lemma 8.3, does not exhibit the CSP when , so is not a universal CSP statistic on . However, trivially refines to the orbit for any . Since refinement is not generally closed under intersections, it is not clear if there is any useful sense in which on words can be “maximally refined.”
It follows from Lemma 8.3 and (5) that Theorem 1.5 is equivalent to the following.
Theorem 8.4**.**
The statistics and are equidistributed modulo on .
Indeed, we were originally led to Theorem 1.5 through an exploration of the irreducible multiplicities of the so-called higher Lie modules (see e.g. [Sch03]), which uncovered the fact that and are equidistributed modulo on . Data exploration led us to conjecture this equidistribution refined to fixed cyclic descent type as in Theorem 8.4. These connections will be explained in a future publication. They also naturally suggest the problem of finding explicit bijections proving Theorem 8.4, which we leave as an open problem.
Problem 8.5**.**
For and any weak composition, find a bijection
[TABLE]
satisfying
[TABLE]
Acknowledgments
We were partially supported by the National Science Foundation grant DMS-1101017. We sincerely thank our advisor, Sara Billey, for her many helpful suggestions, including connections to cyclic sieving, and for her very careful reading of numerous versions of the manuscript. We also thank Vic Reiner for fruitful early discussions related to Lie multiplicities, Yuval Roichman and his collaborators for generously sharing their preprints, and the anonymous referees for their thoughtful comments.
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