Almost partition identities

Abstract

An almost partition identity is an identity for partition numbers that is true asymptotically 100% of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.partitions | identities | asymptotics

On a Partition Identity of Lehmer

“…Remark 1. As proved in [3], the excess in Theorem 1.2 is almost always equal to the number of parts in all self-conjugate partitions of n. Hence, the excess in the number of parts in all partitions in P e (n, 2) over the number of parts in all partitions in P o (n, 2) is almost always equal to the total number of parts in all self-conjugate partitions of n and in all partitions of n into distinct odd parts. More precisely, if N (x) is the number of times the above statement is true for n ≤ x, then lim x→∞ N (x)/x = 1.…”

“…The conjecture was also proved combinatorially by Yang [11] and Ballantine-Bielak [4] independently. This work was followed by generalizations and Beck-type companions to other well known identities (e.g., [3], [5], [9], [11]). In general, a Beck-type companion identity to |P X (n)| = |P Y (n)| is an identity that equates the excess of the number of parts in all partitions in P X (n) over the number of parts in all partitions in P Y (n) to the number of partitions of n satisfying a condition closely related to X (or Y ).…”

“…Equivalently, the excess of Theorem 1.5 equals the number of partitions of n in which, among the parts divisible by r, there is a single even multiple of r and this part is possibly repeated, while all other parts divisible by r are odd multiples of r and they are distinct. Theorem 1.6 below is a restricted Beck-type companion identity to (3), in which we only count the number of parts divisible by r in Q o (n, r), and the number of parts divisible by 2r in P e (n, 2r) and P o (n, 2r). The theorem reduces to Theorem 1.3 when r = 1.…”

Beck-type identities for Euler pairs of order $r$

“…Several additional similar identities were proved in the last two years. See for example [4,7,8,9]. In order to define Beck-type identities, we first introduce the necessary terminology and notation.…”

On Sum-Of-Tails Identities

“…All the coefficients of this series upto q 1000 are in absolute value less than 2, which hints that, possibly, almost all of the coefficients of this series appear infinitely often and most of the terms are zero, similar to that in the case of σ(q). Recently this series also occurred in [4] while studying the generating function of the total number of parts in all self conjugate partitions of a certain integer, where, indeed, the above property is said to follow from [8,Theorem 5]. This motivates us to study this series.…”