On 1212-avoiding restricted growth functions
type: Item
instance of: scholarly article
title: On 1212-avoiding restricted growth functions
zbMATH Open document ID: 1358.05009
author: Zhicong Lin
author: Shishuo Fu
published in: The Electronic Journal of Combinatorics
publication date: 2017-04-10T00:00:00Z
full work available at URL: v24i1p53
review text: Summary: Restricted growth functions (RGFs) avoiding the pattern \(1212\) are in natural bijection with noncrossing partitions. Motivated by recent work of \textit{L. R. Campbell} et al. [``Restricted growth function patterns and statistics'', Preprint, \url{arXiv:1605.04807}], we study five classical statistics bk, ls, lb, rs and rb~on 1212-avoiding RGFs. We show the equidistribution of (ls, rb, lb, bk) and (rb, ls, lb, bk) on 1212-avoiding RGFs by constructing a simple involution. To our surprise, this result was already proved by \textit{R. Simion} [Electron. J. Comb. 7, No.1, Research paper R9, 27 p. (2000; Zbl 0938.05003)] via an involution on noncrossing partitions. Our involution, though turns out essentially the same as Simion's, is defined quite differently and has the advantage that makes the discussion more transparent. Consequently, a multiset-valued extension of Simion's result is discovered. Furthermore, similar approach enables us to prove the equidistribution of (mak, rb, rs, bk) and (rb, mak, rs, bk) on 1212-avoiding RGFs, where ``mak'' is a set partition statistic introduced by Steingrímsson.{ }Through two bijections to Motzkin paths, we also prove that the triple of classical permutation statistics (exc+1, den, inv -- exc) on 321-avoiding permutations is equidistributed with the triple (bk, rb, rs) on 1212-avoiding RGFs, which generalizes another result of Simion. In the course, an interesting \(q\)-analog of the \(\gamma\)-positivity of Narayana polynomials is found.
Mathematics Subject Classification ID: 05A05
zbMATH DE Number: 6703745
zbMATH Keywords: restricted growth function
zbMATH Keywords: pattern avoidance
zbMATH Keywords: noncrossing partitions
zbMATH Keywords: partition statistics
zbMATH Keywords: Narayana polynomials
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6703745