Graphical Mahonian Statistics on Words

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...

Mahonian STAT on words

In 2000, Babson and Steingŕımsson introduced the notion of what is now known as a permutation vincular pattern, and based on it they re-defined known Mahonian statistics and introduced new ones, proving or conjecturing their Mahonity. These conjectures were proved by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006. In 2010, Burstein refined some of these results by giving...

Mahonian Z statistics

Mahonian STAT on rearrangement class of words

In 2000, Babson and Steingrímsson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discoverd by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistributio...

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”,...