Inversion formulae on permutations avoiding 321
type: Item
instance of: scholarly article
title: Inversion formulae on permutations avoiding 321
zbMATH Open document ID: 1326.05004
author: Pingge Chen
author: Zhousheng Mei
author: Suijie Wang
published in: The Electronic Journal of Combinatorics
publication date: 2015-11-26T00:00:00Z
full work available at URL: v22i4p28
review text: Summary: We will study the inversion statistic of 321-avoiding permutations, and obtain that the number of 321-avoiding permutations on \([n]\) with \(m\) inversions is given by \[ |\mathcal S_{n,m}(321)|=\sum_{b\vdash m}\binom{n-\frac{\Delta(b)}{2}}{l(b)}. \] where the sum runs over all compositions \(b=(b_1,b_2,\ldots,b_k)\) of \(m\), i.e., \[ m=b_1+b_2+\cdots+b_k\,\,\text{and}\,\,b_i\geqslant 1, \] \(l(b)=k\) is the length of \(b\), and \(\Delta(b):=|b_1|+|b_2-b_1|+\cdots+|b_k-b_{k-1}|+|b_k|\). We obtain a new bijection from 321-avoiding permutations to Dyck paths which establishes a relation on inversion number of 321-avoiding permutations and valley height of Dyck paths.
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 6513829
zbMATH Keywords: pattern avoidance
zbMATH Keywords: Catalan number
zbMATH Keywords: Dyck path
zbMATH Keywords: generating function
MaRDI profile type: Publication
cites work: A weight-preserving bijection between Schröder paths and Schröder permutations
cites work: Descent sets on 321-avoiding involutions and hook decompositions of partitions
cites work: Refining enumeration schemes to count according to the inversion number
cites work: Some combinatorial properties of Schubert polynomials
cites work: Motzkin paths and reduced decompositions for permutations with forbidden patterns
cites work: Inversion polynomials for 321-avoiding permutations
cites work: Classification of bijections between 321- and 132-avoiding permutations
cites work: Permutation patterns and statistics
cites work: Restricted 3412-avoiding involutions, continued fractions, and Chebyshev polynomials
cites work: A simple and unusual bijection for Dyck paths and its consequences
cites work: Bijections for refined restricted permutations
cites work: Q5585021
cites work: Q4057549
cites work: Permutations with restricted patterns and Dyck paths
cites work: Dyck paths and restricted permutations
cites work: The maximal-inversion statistic and pattern-avoiding permutations
cites work: A generalization of Simion-Schmidt's bijection for restricted permutations
cites work: Q3495358
cites work: On a correspondence between binary trees and a certain type of permutation
cites work: Mahonian pairs
cites work: Restricted permutations
cites work: Generating trees and the Catalan and Schröder numbers
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description: scientific article; zbMATH DE number 6513829