Refined Wilf-equivalences by Comtet statistics

09/09/2020
βˆ™
by   Shishuo Fu, et al.
βˆ™
0
βˆ™

We launch a systematic study of the refined Wilf-equivalences by the statistics π–Όπ—ˆπ—†π—‰ and 𝗂𝖺𝗋, where π–Όπ—ˆπ—†π—‰(Ο€) and 𝗂𝖺𝗋(Ο€) are the number of components and the length of the initial ascending run of a permutation Ο€, respectively. As Comtet was the first one to consider the statistic π–Όπ—ˆπ—†π—‰ in his book Analyse combinatoire, any statistic equidistributed with π–Όπ—ˆπ—†π—‰ over a class of permutations is called by us a Comtet statistic over such class. This work is motivated by a triple equidistribution result of Rubey on 321-avoiding permutations, and a recent result of the first and third authors that 𝗂𝖺𝗋 is a Comtet statistic over separable permutations. Some highlights of our results are: (1) Bijective proofs of the symmetry of the double Comtet distribution (π–Όπ—ˆπ—†π—‰,𝗂𝖺𝗋) over several Catalan and SchrΓΆder classes, preserving the values of the left-to-right maxima. (2) A complete classification of π–Όπ—ˆπ—†π—‰- and 𝗂𝖺𝗋-Wilf-equivalences for length 3 patterns and pairs of length 3 patterns. Calculations of the (π–½π–Ύπ—Œ,𝗂𝖺𝗋,π–Όπ—ˆπ—†π—‰) generating functions over these pattern avoiding classes and separable permutations. (3) A further refinement by the Comtet statistic 𝗂𝖺𝗋, of Wang's recent descent-double descent-Wilf equivalence between separable permutations and (2413,4213)-avoiding permutations.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

βˆ™ 03/18/2021

The equidistribution of some Mahonian statistics over permutations avoiding a pattern of length three

We prove the equidistribution of several multistatistics over some class...
βˆ™ 08/11/2019

Bijective recurrences concerning two SchrΓΆder triangles

Let r(n,k) (resp. s(n,k)) be the number of SchrΓΆder paths (resp. little ...
βˆ™ 10/14/2018

q-Stirling numbers arising from vincular patterns

The distribution of certain Mahonian statistic (called BAST) introduced ...
βˆ™ 02/26/2019

Weighted games of best choice

The game of best choice (also known as the secretary problem) is a model...
βˆ™ 06/05/2019

KAS-term: Extracting Slovene Terms from Doctoral Theses via Supervised Machine Learning

This paper presents a dataset and supervised learning experiments for te...
βˆ™ 10/14/2018

From q-Stirling numbers to the Delta Conjecture: a viewpoint from vincular patterns

The distribution of certain Mahonian statistic (called BAST) introduced ...
βˆ™ 12/22/2019

Catalan words avoiding pairs of length three patterns

Catalan words are particular growth-restricted words counted by the epon...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.