Distribution of permutation statistics across pattern avoidance classes, and the search for a Denert-associated condition equivalent to pattern avoidance

We begin with a discussion of the symmetricity of \maj over \des in pattern avoidance classes, and its relationship to \maj-Wilf equivalence. From this, we explore the distribution of permutation statistics across pattern avoidance for patterns of length 3 and 4. We then begin discussion of Han\u27s bijection, a bijection on permutations which sends the major index to Denert\u27s statistic and the descent number to the (strong) excedance number. We show the existence of several infinite families of fixed points for Han\u27s bijection. Finally, we discuss the image of pattern avoidance classes under Han\u27s bijection, for the purpose of finding a condition which has the same distribution of \den over \exc as pattern avoidance does of \maj over \des

Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting

Type A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group S_n are different, they both satisfy a convolution property. Strong evidence is given that when the underlying parameter $q$ satisfies $gcd(n,q-1)=1$, the induced measures on conjugacy classes of the symmetric group coincide. This gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents. It is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness. Generating functions for the first pile size in patience sorting from decks with repeated values are derived. This relates to random matrices.Comment: Galley version for J. Alg.; minor revisions in Sec.

Permutation patterns and statistics

Let S_n denote the symmetric group of all permutations of the set {1, 2, ...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of Pi in the sense of pattern avoidance. One of the celebrated notions in pattern theory is that of Wilf-equivalence, where Pi and Pi' are Wilf equivalent if #Av_n(Pi)=#Av_n(Pi') for all n\ge0. In a recent paper, Sagan and Savage proposed studying a q-analogue of this concept defined as follows. Suppose st:S->N is a permutation statistic where N represents the nonnegative integers. Consider the corresponding generating function, F_n^{st}(Pi;q) = sum_{sigma in Av_n(Pi)} q^{st sigma}, and call Pi,Pi' st-Wilf equivalent if F_n^{st}(Pi;q)=F_n^{st}(Pi';q) for all n\ge0. We present the first in-depth study of this concept for the inv and maj statistics. In particular, we determine all inv- and maj-Wilf equivalences for any Pi containd in S_3. This leads us to consider various q-analogues of the Catalan numbers, Fibonacci numbers, triangular numbers, and powers of two. Our proof techniques use lattice paths, integer partitions, and Foata's fundamental bijection. We also answer a question about Mahonian pairs raised in the Sagan-Savage article.Comment: 28 pages, 5 figures, tightened up the exposition, noted that some of the conjectures have been prove

Signed Mahonians

A classical result of MacMahon gives a simple product formula for the generating function of major index over the symmetric group. A similar factorial-type product formula for the generating function of major index together with sign was given by Gessel and Simion. Several extensions are given in this paper, including a recurrence formula, a specialization at roots of unity and type $B$ analogues.Comment: 23 page

Equidistribution of negative statistics and quotients of Coxeter groups of type B and D

We generalize some identities and q-identities previously known for the symmetric group to Coxeter groups of type B and D. The extended results include theorems of Foata and Sch\"utzenberger, Gessel, and Roselle on various distributions of inversion number, major index, and descent number. In order to show our results we provide caracterizations of the systems of minimal coset representatives of Coxeter groups of type B and D.Comment: 18 pages, 2 figure