Upper bounds for the Stanley–Wilf limit of 1324 and other layered patterns

Abstract

We prove that the Stanley-Wilf limit of any layered permutation pattern of length l is at most 4l(2), 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. We also conjecture that, for any k >= 0, the set of 1324-avoiding permutations with k inversions contains at least as many permutations of length n + 1 as those of length n. We show that if this is true then the Stanley-Wilf limit for 1324 is at most e(pi root 2/3) similar or equal to 13.001954

Staircases, dominoes, and the growth rate of 1324-avoiders

“…The characterisation is illustrated in Figure 1(a). Note that this characterisation is consistent with the decomposition due to Claesson, Jelínek and Steingrímsson [12], since the cells that avoid 132 (respectively, 213) collectively also avoid 132 (resp., 213). Furthermore, momentarily dropping the 1324-condition, the class comprising all permutations that lie in the staircase illustrated in Figure 1(a) has growth rate equal to 16, which can be deduced immediately from [2,Theorem 3].…”

“…2004: Bóna [6] 288 2005: Bóna [7] 9 2006: Albert et al [1] 9.47 2012: Claesson, Jelínek and Steingrímsson [12] 16 2014: Bóna [8] 13.93 2015: Bóna [9] 13.74 2015: Bevan [4] 9.81 This article 10.27 13.5 and final section, we take multiple copies of this domino class and 'glue' them together in ways that are consistent with the structural characterisation, and thus establish the claimed bounds on the growth rate of Av(1324).…”

“…The following structural characterisation has implicitly been used in the literature (notably the red and blue subtrees of Bevan [4], and the colouring approaches taken by [8,9,12] to express Av(1324) as the merge of a permutation in Av(132) with a permutation in Av(213)), but it does not seem to have been stated explicitly before now. The characterisation is illustrated in Figure 1(a).…”

“…In addition to these, Claesson, Jelínek and Steingrímsson [12] make a conjecture regarding the number of permutations with a fixed number of inversions of each length, which if resolved would give an improved upper bound of e π √ 2/3 ≈ 13.001954. Our contribution to the growth rate study of Av (1324) is to provide new lower and upper bounds on gr(Av(1324)).…”

Permutations avoiding 1324 and patterns in Łukasiewicz paths

“…The last few years have seen a steady reduction in upper bounds on the growth rate, based on a colouring scheme of Claesson, Jelínek & Steingrímsson [5] which yields a value of 16. Bóna [3] has now reduced this to 13.73718 by employing a refined counting argument.…”