Resolution of a Conjecture of Andrews and Lewis Involving Cranks of Partitions
نویسندگان
چکیده
In [1] Andrews and Lewis conjecture that the sign of the number of partitions of n with crank congruent to 0 mod 3, minus the number of partitions of n with crank congruent to 1 mod 3 is determined by the congruence class of n mod 3 apart from a finite number of specific exceptions. We prove this by using the “Circle Method” to approximate the value of this difference to great enough accuracy to determine its sign for all sufficiently large n.
منابع مشابه
A Proof of Andrews’ Conjecture on Partitions with No Short Sequences
Our main result establishes Andrews’ conjecture for the asymptotic of the generating function for the number of integer partitions of n without k consecutive parts. The methods we develop are applicable in obtaining asymptotics for stochastic processes that avoid patterns, as a result they yield asymptotics for the number of partitions that avoid patterns. Holroyd, Liggett, and Romik, in connec...
متن کاملElementary Proof of MacMahon’s Conjecture
Major Percy A. MacMahon’s first paper on plane partitions [4] included a conjectured generating function for symmetric plane partitions. This conjecture was proven almost simultaneously by George Andrews and Ian Macdonald, Andrews using the machinery of basic hypergeometric series [1] and Macdonald employing his knowledge of symmetric functions [3]. The purpose of this paper is to simplify Macd...
متن کاملCranks and T -cores
New statistics on partitions (called cranks) are defined which combinatorially prove Ramanujan’s congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the equinumerous crank classes. The cranks are closely related to the t-core of a partition. Using q-series, some explicit formulas are given for the number of partitions which are t-cores. Some related...
متن کاملA PROOF OF GEORGE ANDREWS’ AND DAVID ROBBINS’ q-TSPP CONJECTURE
The conjecture that the orbit-counting generating function for totally symmetric plane partitions can be written as an explicit product-formula, has been stated independently by George Andrews and David Robbins around 1983. We present a proof of this long-standing conjecture.
متن کاملRelations between the Rank and the Crank
New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.
متن کامل