39 research outputs found
An involution on \beta(1,0)-trees
In [Decompositions and statistics for \beta(1,0)-trees and nonseparable
permutations, Advances Appl. Math. 42 (2009) 313--328] we introduced an
involution, h, on \beta(1,0)-trees. We neglected, however, to prove that h
indeed is an involution. In this note we provide the missing proof. We also
refine an equidistribution result given in the same paper.Comment: arXiv admin note: text overlap with arXiv:0801.403
The Möbius function of separable and decomposable permutations
We give a recursive formula for the Moebius function of an interval in the poset of permutations ordered by pattern containment in the case where is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1, 2, ..., k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Moebius function in the case where and are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. A consequence of the formula is that the Moebius function of such an interval is bounded by the number of occurrences of as a pattern in . We also show that for any separable permutation the Moebius function of is either 0, 1 or -1
Sorting with pattern-avoiding stacks : the 132-machine
This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari (2020). These devices consist of two stacks, through which a permutation is passed in order to sort it, where the content of each stack must at all times avoid a certain pattern. Here we characterize and enumerate the set of permutations that can be sorted when the first stack is 132-avoiding, solving one of the open problems proposed by the above mentioned authors. To that end we present several connections with other well known combinatorial objects, such as lattice paths and restricted growth functions (which encode set partitions). We also provide new proofs for the enumeration of some sets of pattern-avoiding restricted growth functions and we expect that the tools introduced can be fruitfully employed to get further similar results
Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns
We prove that the Stanley-Wilf limit of any layered permutation pattern of
length is at most , and that the Stanley-Wilf limit of the
pattern 1324 is at most 16. These bounds follow from a more general result
showing that a permutation avoiding a pattern of a special form is a merge of
two permutations, each of which avoids a smaller pattern. If the conjecture is
true that the maximum Stanley-Wilf limit for patterns of length is
attained by a layered pattern then this implies an upper bound of for
the Stanley-Wilf limit of any pattern of length .
We also conjecture that, for any , the set of 1324-avoiding
permutations with inversions contains at least as many permutations of
length as those of length . We show that if this is true then the
Stanley-Wilf limit for 1324 is at most
Relatório de estágio em farmácia comunitária
Relatório de estágio realizado no âmbito do Mestrado Integrado em Ciências Farmacêuticas, apresentado à Faculdade de Farmácia da Universidade de Coimbr
Permutations statistics of indexed and poset permutations
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1992.Includes bibliographical references (p. 44-45).by Einar Steingrímsson.Ph.D
Statistics on ordered partitions of sets 1
The Stirling numbers of the second kind, S(n, k), which count the partitions of an n-element set into k blocks, have been much studied. Their q-analog Sq(n, k), the q-Stirling numbers of the second kind, can be defined by Sq(n, k) = 0 if k> n or k < 0 and, for n ≥ k ≥ 0, by the identit