Major Index Statistic Research Articles

The birational Lalanne–Kreweras involution

The Lalanne–Kreweras involution is an involution on the set of Dyck paths which combinatorially exhibits the symmetry of the number of valleys and major index statistics. We define piecewise-linear and birational extensions of the Lalanne–Kreweras involution. Actually, we show that the Lalanne–Kreweras involution is a special case of a more general operator, called rowvacuation, which acts on the antichains of any graded poset. Rowvacuation, like the closely related and more studied rowmotion operator, is a composition of toggles. We obtain the piecewise-linear and birational lifts of the Lalanne–Kreweras involution by using the piecewise-linear and birational toggles of Einstein and Propp. We show that the symmetry properties of the Lalanne–Kreweras involution extend to these piecewise-linear and birational lifts.

Cyclic Sieving for cyclic codes

Prompted by a question of Jim Propp, this paper examines the cyclic sieving phenomenon (CSP) in certain cyclic codes. For example, it is shown that, among dual Hamming codes over Fq, the generating function for codedwords according to the major index statistic (resp. the inversion statistic) gives rise to a CSP when q=2 or q=3 (resp. when q=2). A byproduct is a curious characterization of the irreducible polynomials in F2[x] and F3[x] that are primitive.

Odd and Even Major Indices and One-Dimensional Characters for Classical Weyl Groups

We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz’s identity, of the Gessel–Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs.

Sequences, q-multinomial Identities, Integer Partitions with Kinds, and Generalized Galois Numbers

Using sequences of finite length with positive integer entries and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their q-analog form via combinatorial proofs. Using the major index statistic on sequences, a connection between finite differences of the coefficients of generalized Galois numbers and integer partitions with kinds is established.

Tableau posets and the fake degrees of coinvariant algebras

We introduce two new partial orders on the standard Young tableaux of a given partition shape, in analogy with the strong and weak Bruhat orders on permutations. Both posets are ranked by the major index statistic offset by a fixed shift. The existence of such ranked poset structures allows us to classify the realizable major index statistics on standard tableaux of arbitrary straight shape and certain skew shapes. By a theorem of Lusztig–Stanley, this classification can be interpreted as determining which irreducible representations of the symmetric group exist in which homogeneous components of the corresponding coinvariant algebra, strengthening a recent result of the third author for the modular major index. Our approach is to identify patterns in standard tableaux that allow one to mutate descent sets in a controlled manner. By work of Lusztig and Stembridge, the arguments extend to a classification of all nonzero fake degrees of coinvariant algebras for finite complex reflection groups in the infinite family of Shephard–Todd groups.

Refined cyclic sieving on words for the major index statistic

Reiner–Stanton–White (Reiner et al., 2004) 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 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 (Wagon and Wilf, 1994).

A Simple Transformation for Mahonian Statistics on Labelings of Rake Posets

We present a simple transformation for the inversion number and major index statistics on the labelings of a rooted tree with n vertices in the form of a rake with k teeth. The special case $$k=0$$ provides a simple transformation for the Mahonian statistics on the set $$\mathfrak {S}_n$$ of permutations of $$\{1,2,\dots ,n\}$$ . We also extend the transformation to a bijective interpretation of the fact that the major index of the equivalence classes of the labelings is equidistributed with the major index of the permutations in $$\mathfrak {S}_n$$ satisfying the condition that the elements $$1,2,\dots ,k$$ appear in increasing order.

Euler–Mahonian statistics and descent bases for semigroup algebras

We consider quotients of the unit cube semigroup algebra by particular Zr≀Sn-invariant ideals. Using Gröbner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations (π,ϵ)∈Zr≀Sn and each element encodes the negative descent and negative major index statistics on (π,ϵ). This gives an algebraic interpretation of these statistics that was previously unknown. This basis of the Zr≀Sn-quotients allows us to recover certain combinatorial identities involving Euler–Mahonian distributions of statistics.

A minimaj-preserving crystal on ordered multiset partitions

We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization Rn,k due to Haglund, Rhoades and Shimozono of the coinvariant algebra Rn. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.

A new q-Selberg integral, Schur functions, and Young books

Recently, Kim and Oh expressed the Selberg integral in terms of the number of Young books which are a generalization of standard Young tableaux of shifted staircase shape. In this paper the generating function for Young books according to major index statistic is considered. It is shown that this generating function can be written as a Jackson integral which gives a new q-Selberg integral. It is also shown that the new q-Selberg integral has an expression in terms of Schur functions.

Results and conjectures on simultaneous core partitions

An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects.In particular, we prove that 2n- and (2mn+1)-core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type Cn, generalizing a result of Fishel–Vazirani for type A. We also introduce a major index statistic on simultaneous n- and (n+1)-core partitions and on self-conjugate simultaneous 2n- and (2n+1)-core partitions that yield q-analogs of the Coxeter–Catalan numbers of type A and type C.We present related conjectures and open questions on the average size of a simultaneous core partition, q-analogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q,t-Catalan numbers.

A refinement of Wilf-equivalence for patterns of length 4

In their paper [6], Dokos et al. conjecture that the major index statistic is equidistributed among 1423-avoiding, 2413-avoiding, and 3214-avoiding permutations. In this paper we confirm this conjecture by constructing two major index preserving bijections, Θ:Sn(1423)→Sn(2413) and Ω:Sn(3214)→Sn(2413). In fact, we show that Θ (respectively, Ω) preserves numerous other statistics including the descent set, right-to-left maxima (respectively, left-to-right minima), and a statistic we call steps. Additionally, Θ (respectively, Ω) fixes all permutations avoiding both 1423 and 2413 (respectively, 3214 and 2413).

Euler–Mahonian statistics via polyhedral geometry

A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to a pair of such statistics is an Euler–Mahonian distribution, a bivariate polynomial encoding the statistics; such distributions often appear in rational bivariate generating-function identities. We use techniques from polyhedral geometry to establish new multivariate identities generalizing those giving rise to many of the known Euler–Mahonian distributions. The original bivariate identities are then specializations of these multivariate identities. As a consequence of these new techniques we obtain bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics.

A multivariate “inv” hook formula for forests

Bjorner and Wachs provided two q-generalizations of Knuth’s hook formula counting linear extensions of forests: one involving the major index statistic, and one involving the inversion number statistic. We prove a multivariate generalization of their inversion number result, motivated by specializations related to the modular invariant theory of finite general linear groups.

Recursions and divisibility properties for combinatorial Macdonald polynomials

Combinatorics For each integer partition mu, let e (F) over tilde (mu)(q; t) be the coefficient of x(1) ... x(n) in the modified Macdonald polynomial (H) over tilde (mu). The polynomial (F) over tilde (mu)(q; t) can be regarded as the Hilbert series of a certain doubly-graded S(n)-module M(mu), or as a q, t-analogue of n! based on permutation statistics inv(mu) and maj(mu) that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of (F) over tilde (mu) to prove some recursions characterizing these polynomials, and other related ones, when mu is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all mu, we show that e (F) over tilde (mu)(q, t) is divisible by certain q-factorials and t-factorials depending on mu. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express e (F) over tilde ((2n)) (q, t) as a sum of q, t-analogues of n!2(n) indexed by perfect matchings.

Major index for 01-fillings of moon polyominoes

We propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F ( M , s ; A ) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distribution as the number of north–east chains, which are the natural extension of inversions (resp. crossings) for permutations (resp. set partitions). Hence our result generalizes the classical equidistribution results for the permutation statistics inv and maj. Two proofs are presented. The first is an algebraic one using generating functions, and the second is a bijection on 01-fillings of moon polyominoes in the spirit of Foata's second fundamental transformation on words and permutations.

Descents, inversions, and major indices in permutation groups

A multivariate generating function involving the descent, major index, and inversion statistic first given by Ira Gessel is generalized to other permutation groups. We provide generating functions for variants of these three statistics for the Weyl groups of type B and D, wreath product groups, and multiples of permutations. All of our ideas are combinatorial in nature and exploit fundamental relationships between the elementary and homogeneous symmetric functions.

Combinatorial proofs of bivariate generating functions on Coxeter groups

McMahon’s result that states the length and major index statistics are equidistributed on the symmetric group Sn has generalizations to other Coxeter groups. Adin, Brenti and Roichman have defined analogues of those statistics for the hyperoctahedral group, Bn and proved equidistribution theorems. Using combinatorial interpretations of Regev and Roichman’s statistics length and delent, we consider a specialization of the bimahonian generating function for the alternating group An. We define a new delent statistic, delL, for the alternating subgroup of the hyperoctahedral group, Ln ⊂ Bn and give a combinatorial proof of the bimahonian generating function ∑ π∈Ln q lL(π)tdelL(π). Mathematics Subject Classification: 20F55, 05A15

Euler–Mahonian polynomials for [formula omitted

The L-reverse major index statistic, rmaj L , is defined on the group of colored permutations, C a ≀ S n , using the < L order and the L-descent number statistic, des L , which were recently introduced by Regev and Roichman. The distribution of des L and the bi-statistic ( des L , rmaj L ) is studied, yielding new wreath-product analogues of the Eulerian and q-Euler–Mahonian polynomials, and a generalization of Carlitz's identity.

M-Rook numbers and a generalization of a formula of Frobenius to [formula omitted

In ordinary rook theory, rook placements are associated to permutations of the symmetric group S n . We provide a generalization of this theory in which “ m-rook placements” are related to elements of C m ≀ S n , where C m is the cyclic group of order m. Within this model, we define and interpret combinatorially a p , q -analogue of the m-rook numbers. We also define a p , q -analogue of the m-hit numbers and show that the coefficients of these polynomials in p and q are nonnegative integers for m-Ferrers boards. Finally, we define statistics des m ( σ ) , maj m ( σ ) , and comaj m ( σ ) as analogues of the ordinary descent, major, and comajor index statistics and prove a generalization of a formula of Frobenius that relates these statistics to generalized p , q -Stirling numbers of the second kind.