Mahonian Z statistics

Mahonian Z statistics

Graphical Mahonian Statistics on Words

Foata and Zeilberger defined the graphical major index, majU , and the graphical inversion index, invU , for words over the alphabet {1, 2, . . . , n}. These statistics are a generalization of the classical permutation statistics maj and inv indexed by directed graphs U . They showed that majU and invU are equidistributed over all rearrangement classes if and only if U is bipartitional. In this...

Fix-euler-mahonian Statistics on Wreath Products

In 1997 Clarke, Han, and Zeng introduced a q-analogue of Euler’s difference table for n! using a key bijection Ψ on symmetric groups. In this paper we extend their results to the wreath product of a cyclic group with the symmetric group. By generalizing their bijection Ψ we prove the equidistribution of the triple statistics (fix, exc, fmaj) and (fix, exc, fmaf) on wreath products, where “fix”,...

Equidistributions of Mahonian Statistics over Pattern Avoiding

A Mahonian d-function is a Mahonian statistic that can be expressed as a linear combination of vincular pattern statistics of length at most d. Babson and Steingŕımsson [2] classified all Mahonian 3-functions up to trivial bijections (see Table 1) and identified many of them with well-known Mahonian statistics in the literature. In [1] we prove a host of Mahonian 3-function equidistributions ov...

Euler-mahonian Statistics on Ordered Partitions

An ordered partition with k blocks of [n] := {1, 2, . . . , n} is a sequence of k disjoint and nonempty subsets, called blocks, whose union is [n]. Clearly the number of such ordered partitions is k!S(n, k), where S(n, k) is the Stirling number of the second kind. A statistic on ordered partitions of [n] with k blocks is called Euler-Mahonian statistics if its generating polynomial is [k]q!Sq(n...