Mahonian STAT on words

Sergey Kitaev, Vincent Vajnovszki

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

In 2000, Babson and Steingr\'imsson 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 a bijection between permutations with a fixed value for the major index and those with the same value for STAT, where STAT is one of the statistics defined and proved to be Mahonian in the 2000 Babson and Steingr\'imsson's paper. Several other statistics are preserved as well by Burstein's bijection. At the Formal Power Series and Algebraic Combinatorics Conference (FPSAC) in 2010, Burstein asked whether his bijection has other interesting properties. In this paper, we not only show that Burstein's bijection preserves the Eulerian statistic ides, but also use this fact, along with the bijection itself, to prove Mahonity of the statistic STAT on words we introduce in this paper. The words statistic STAT introduced by us here addresses a natural question on existence of a Mahonian words analogue of STAT on permutations. While proving Mahonity of our STAT on words, we prove a more general joint equidistribution result involving two six-tuples of statistics on (dense) words, where Burstein's bijection plays an important role.
LanguageEnglish
Pages157-162
JournalInformation Processing Letters
Volume116
Early online date2016
DOIs
Publication statusE-pub ahead of print - 2016

Fingerprint

Bijection
Statistics
Statistic
Permutation
Major Index
Equidistribution
Formal Power Series
Combinatorics
Analogue

Keywords

  • Mahonian statistics
  • Burstein's bijection
  • vincular pattern

Cite this

Kitaev, Sergey ; Vajnovszki, Vincent. / Mahonian STAT on words. In: Information Processing Letters. 2016 ; Vol. 116. pp. 157-162.
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abstract = "In 2000, Babson and Steingr\'imsson 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 a bijection between permutations with a fixed value for the major index and those with the same value for STAT, where STAT is one of the statistics defined and proved to be Mahonian in the 2000 Babson and Steingr\'imsson's paper. Several other statistics are preserved as well by Burstein's bijection. At the Formal Power Series and Algebraic Combinatorics Conference (FPSAC) in 2010, Burstein asked whether his bijection has other interesting properties. In this paper, we not only show that Burstein's bijection preserves the Eulerian statistic ides, but also use this fact, along with the bijection itself, to prove Mahonity of the statistic STAT on words we introduce in this paper. The words statistic STAT introduced by us here addresses a natural question on existence of a Mahonian words analogue of STAT on permutations. While proving Mahonity of our STAT on words, we prove a more general joint equidistribution result involving two six-tuples of statistics on (dense) words, where Burstein's bijection plays an important role.",
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Mahonian STAT on words. / Kitaev, Sergey; Vajnovszki, Vincent.

In: Information Processing Letters, Vol. 116, 2016, p. 157-162.

Research output: Contribution to journalArticle

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AU - Kitaev, Sergey

AU - Vajnovszki, Vincent

PY - 2016

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AB - In 2000, Babson and Steingr\'imsson 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 a bijection between permutations with a fixed value for the major index and those with the same value for STAT, where STAT is one of the statistics defined and proved to be Mahonian in the 2000 Babson and Steingr\'imsson's paper. Several other statistics are preserved as well by Burstein's bijection. At the Formal Power Series and Algebraic Combinatorics Conference (FPSAC) in 2010, Burstein asked whether his bijection has other interesting properties. In this paper, we not only show that Burstein's bijection preserves the Eulerian statistic ides, but also use this fact, along with the bijection itself, to prove Mahonity of the statistic STAT on words we introduce in this paper. The words statistic STAT introduced by us here addresses a natural question on existence of a Mahonian words analogue of STAT on permutations. While proving Mahonity of our STAT on words, we prove a more general joint equidistribution result involving two six-tuples of statistics on (dense) words, where Burstein's bijection plays an important role.

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