On 1212-avoiding restricted growth functions
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.
scientific article; zbMATH DE number 6703745
scientific article