3 research outputs found
Equidistributions of Mahonian statistics over pattern avoiding permutations
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 Steingrimsson classified all Mahonian 3-functions up to trivial bijections
and identified many of them with well-known Mahonian statistics in the
literature. We prove a host of Mahonian 3-function equidistributions over
pattern avoiding sets of permutations. Tools used include block decomposition,
Dyck paths and generating functions.Comment: 31 pages, 4 figures, 2 table
Mahonian STAT on rearrangement class of words
In 2000, Babson and Steingr\'{i}msson 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
equidistribution result involving two sextuple statistics on the whole set of
words with fixed length and alphabet. Moreover, their computer experiments
hinted at a finer involution on , the rearrangement class of a given word
. We construct such an involution in this paper, which yields a comparable
joint equidistribution between two sextuple statistics over . Our
involution builds on Burstein's involution and Foata-Sch\"{u}tzenberger's
involution that utilizes the celebrated RSK algorithm.Comment: 11 page
From -Stirling numbers to the Delta Conjecture: a viewpoint from vincular patterns
The distribution of certain Mahonian statistic (called )
introduced by Babson and Steingr\'{i}msson over the set of permutations that
avoid vincular pattern , is shown bijectively to match the
distribution of major index over the same set. This new layer of
equidistribution is then applied to give alternative interpretations of two
related -Stirling numbers of the second kind, studied by Carlitz and Gould.
Moreover, extensions to an Euler-Mahonian statistic over ordered set
partitions, and to statistics over ordered multiset partitions present
themselves naturally. The latter of which is shown to be related to the
recently proven Delta Conjecture. During the course, a refined relation between
and its reverse complement is derived as well.Comment: 29 pages, a new extension to ordered multiset partitions adde