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Lecture Hall Partitions and the Wreath Products Ck Sn

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Lecture Hall Partitions and the Wreath Products Ck Sn Lecture hall partitions and the wreath products Ck o Sn Thomas W. Pensyl and Carla D. Savage Department of Computer Science, North Carolina State University Raleigh, NC 27695-8206, USA [email protected] and [email protected] submitted: January 31, 2012; revised: December 12, 2012; February 26, 2013 Abstract It is shown that statistics on the wreath product groups, Ck o Sn, can be interpreted in terms of natural statistics on lecture hall partitions. Lecture hall theory is applied to prove distribution results for statistics on Ck oSn. Finally, some new statistics on Ck oSn are introduced, inspired by lecture hall theory, and their distributions are derived. 1 Introduction The purpose of this note is to show that statistics on the wreath product Ck o Sn of a cyclic group Ck, of order k, and the symmetric group Sn, can be interpreted in terms of natural statistics on lecture hall partitions. We demonstrate that lecture hall theory can be used to prove results about the distribution of statistics on Ck o Sn. We introduce some new statistics on Ck o Sn, inspired by lecture hall partitions, including a quadratic version of “flag-major index", and prove distribution results for these statistics. The paper is organized as follows. In Section 2, we define the s-lecture hall partitions and state a few useful results. Section 3 is devoted to statistics of interest on the wreath product groups and a very brief discussion of what is known. Section 4 introduces s- inversion sequences, which will be used to relate statistics on Ck o Sn to statistics on lecture hall partitions. Section 5 describes a bijection between (k; 2k; : : : ; nk)-inversion sequences and Ck o Sn that allows statistics to be translated from one domain to another. Section 6 reviews recent work of Savage-Schuster [13] relating inversion sequences to lecture hall partitions. This work was developed with the intention of extending work on permutation statistics to a more general setting. 1 Section 7 is the heart of the paper. We prove there a theorem which allows us to apply the tools of Section 6 to Ck o Sn. This contains our main results relating statistics such as descent, flag-major index and flag-inversion number to statistics on lecture hall partitions, also proving an Euler-Mahonian distribution result. In Section 8 we define a new statistic \lhall " on Ck o Sn and derive its surprisingly nice distribution. In Section 9, we are led to define a distorted version of the descent statistic on Ck o Sn, that reveals an even closer connection to lecture hall partitions. A few words about notation: Z is the set of integers, R the set of real numbers, Sn the n set of permutations of n elements; [ j ] = f1; 2; : : : ; jg, where [ 0 ] = ;;[ n ]q = (1−q )=(1−q); and for x = (x1; x2; : : : ; xn), jxj = x1 + x2 + ··· + xn. 2 Lecture hall partitions For a sequence s = fsigi≥1 of positive integers, the s-lecture hall partitions are the elements of the set (s) n λ1 λ2 λn Ln = λ 2 Z 0 ≤ ≤ ≤ · · · ≤ : s1 s2 sn (1;2;3;4) (1;3;5;7) For example, (0; 1; 3; 4) 2 Ln , but (0; 1; 3; 4) 62 Ln , since 3=5 > 4=7. (1;2;:::;n) The original lecture hall partitions Ln = Ln were introduced by Bousquet-M´elou and Eriksson in [3], where they showed that n X Y 1 yjλj = : (1) 1 − y2i−1 λ2Ln i=1 In [4] they proved the following refinement, which will be useful in the present work. Theorem 1. The Refined Lecture Hall Theorem [4]: For any nonnegative integer n, n X Y 1 + qyi qjdλejyjλj = ; (2) 1 − q2yn+i λ2Ln i=1 where dλe = (dλ1=1e ; dλ2=2e ;:::; dλn=ne). If the largest part in a lecture hall partition in Ln is constrained, we have the following. Theorem 2. [8, 13] For integers n ≥ 1 and t ≥ 0, X jdλej n q = [ t + 1 ]q : (3) λ2Ln; λn≤tn 2 For example, when n = 3 and t = 1, the set fλ 2 L3 j λ3 ≤ 3g has the eight elements: f(0; 0; 0); (0; 0; 1); (0; 0; 2); (0; 0; 3); (0; 1; 2); (0; 1; 3); (0; 2; 3); (1; 2; 3)g and X dλ1=1e+dλ2=2e+dλ3=3e 2 3 3 q = 1 + 3q + 3q + q = [ 2 ]q : λ2L3; λ3≤3 3 Statistics on Ck o Sn An element π 2 Sn is a bijection π :[ n ] ! [ n ] and we write π = (π1; : : : ; πn), to mean that π(i) = πi.A descent in π 2 Sn is a position i 2 [ n − 1 ] such that πi > πi+1. The set of all descents of π is Des π and des π = jDes πj. The inversion number of π is inv π = jf(i; j) j 1 ≤ i < j ≤ n and πi > πjgj: For example, if π = (5; 4; 1; 3; 2), then Des π = f1; 2; 4g, des π = 3 and inv π = 8. For positive integers k and n, we view Ck o Sn combinatorially as a set of pairs (π; σ): n Ck o Sn = f(π; σ) j π 2 Sn; σ 2 f0; 1; : : : ; k − 1g g: We use the notation πσ to denote (π; σ) and write σ σ1 σ2 σn π = (π1 ; π2 ; : : : ; πn ) = ((π1; : : : ; πn); (σ1; : : : ; σn)) = (π; σ): Statistics on Ck o Sn (or k-colored permutations or k-indexed permutations) have been studied by many, starting with Reiner's work on signed permutations [12], followed by independent work of Brenti [5] and Steingr´ımsson[14] on the more general wreath products. Pairs of \(descent, major index)" statistics have been found, satisfying relations like Carlitz's q-Eulerian polynomials, starting with work of Adin, Brenti, and Roichman [1]. There have very recently been many new and exciting discoveries, including [7, 10, 9, 2]. It is remarkable the many variations in the definitions of the statistics, even when they give the same distribution. σ We start with a fairly standard definition of descent. The descent set of π 2 Ck o Sn is σ Des π = fi 2 f0; 1; : : : ; n − 1g j σi < σi+1; or σi = σi+1 and πi > πi+1g; (4) with the convention that π0 = σ0 = 0. 3 We will consider the following statistics defined on Ck o Sn. des πσ = jDes πσj X comaj πσ = (n − i) i2Des πσ n σ σ X fmaj π = k comaj π − σi i=1 n σ X finv π = inv π + iσi: i=1 σ 1 1 0 0 2 σ σ As an example, for π = (5 ; 4 ; 1 ; 3 ; 2 ) 2 C3 o S5, we have Des π = f0; 1; 4g; des π = 3; comaj πσ = 10; fmaj πσ = 26; and finv πσ = 21. Note that this definition of fmaj differs a bit from those appearing elsewhere, even among those who define the descent set as in (4) ([1, 7]). Using lecture hall theory, we will show, among other things: X σ X σ qfmaj π = qfinv π ; (5) σ σ π 2CkoSn π 2CkoSn P fmaj πσ des πσ X πσ2C oS q x [ kt + 1 ]n xt = k n ; (6) q Qn (1 − xqki) t≥0 i=0 P fmaj πσ des πσ σ q x X jdλej dλn=(kn)e π 2CkoSn q x = Qn ki : (7) i=1(1 − xq ) λ2Ln Relations of the form (6), for general k, have been found only recently, starting with Chow and Mansour [7] and Hyatt [10], sometimes with slightly different definitions of Des or fmaj . Our intention here is to highlight our methods, which are quite novel, and which allow us to prove new results like (7). 4 Statistics on s-inversion sequences The connection between statistics on Ck o Sn and statistics on lecture hall partitions will be made via statistics on inversion sequences. (s) Given a sequence s = fsigi≥1 of positive integers, and positive integer n, the set In of s-inversion sequences is defined by (s) n In = f(e1; : : : ; en) 2 Z j 0 ≤ ei < si for 1 ≤ i ≤ ng : (s) The familiar \inversion sequences" associated with permutations are the elements of In for s = (1; 2; : : : ; n). 4 (s) The ascent set of an inversion sequence e 2 In is the set e e Asc e = i 2 f0; 1; : : : ; n − 1g i < i+1 ; si si+1 (3;6;9;12;15) with the convention that e0 = 0. For example, as an element of I5 , the inversion sequence e = (1; 3; 2; 2; 13) has the ascent set Asc e = f0; 1; 4g. (s) The following statistics on In were defined in [13]: asc e = Asc e ; X amaj e = (n − i); i2Asc e n X jej = ei; i=1 X lhp e = −|ej + (si+1 + ::: + sn) : i2Asc e (3;6;9;12;15) For e = (1; 3; 2; 2; 13) 2 I5 , we have asc e = 3; amaj e = 10; jej = 21; and lhp e = 81. In this paper, our focus is the sequence s = (k; 2k; : : : ; nk), where k is a positive integer. (k;2k;:::;nk) Let In;k = In . We will require two new statistics on In;k: n X ej N(e) = ; j j=1 Ifmaj e = k amaj e − N(e): (3;6;9;12;15) For e = (1; 3; 2; 2; 13) 2 I5 , N(e) = 4 and Ifmaj e = 26. 5 From statistics on Ck o Sn to statistics on In;k We will make use of the following bijection between Sn and In;1 which was proved in [13] to have the required properties.
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