Integrals, partitions, and cellular automata

Abstract

Abstract. We prove thatwhere f (x) is the decreasing function that satisfiesWhen a is an integer and b = a + 1 we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having a consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.

On the Andrews–Zagier asymptotics for partitions without sequences

“…Interestingly, the above described probability model also appears in the study of integer partitions [4,8]. In particular,…”

“…Holroyd, Liggett, and Romik [8] introduced the following probability models: Let 0 < s < 1 and C 1 , C 2 , · · · be independent events with probabilities P s (C n ) := 1 − e −ns under a certain probability measure P s . Let A k be the event…”

“…The asymptotic of [8] was improved by Mahlburg and the first author [6] log(g k (e −s )). Finally, Kane and the third author [9], using a technique similar to the transfer matrix method of statistical mechanics, proved Conjecture 1.1 with C k = √ 2π/k.…”

Partitions with short sequences and mock theta functions

“…Note that this is a much stronger result than the k ϭ 2 instance of theorem 2 of ref. 5. We hope that these initial successes will lead to an effort to understand the many implications of Theorems 1 and 2.…”

“…Indeed, in ref. 5, the more general partition function p k (n) is considered; p k (n) is the number of partitions of n that do not contain any sequence of consecutive integers of length k. Thus, from our previous calculation we see that p 2 (7) ϭ 8. Holroyd et al also study the related generating function:…”