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

Abstract

Partition identities are often statements asserting that the set P X of partitions of n subject to condition X is equinumerous to the set P Y of partitions of n subject to condition Y . A Beck-type identity is a companion identity to |P X | = |P Y | asserting that the difference b(n) between the number of parts in all partitions in P X and the number of parts in all partitions in P Y equals a c|P X ′ | and also c|P Y ′ |, where c is some constant related to the original identity, and X ′ , respectively Y ′ , is a condition on partitions that is a very slight relaxation of condition X, respectively Y . A second Beck-type identity involves the difference b ′ (n) between the total number of different parts in all partitions in P X and the total number of different parts in all partitions in P Y . We extend these results to Beck-type identities accompanying all identities given by Euler pairs of order r (for any r ≥ 2). As a consequence, we obtain many families of new Beck-type identities. We give analytic and bijective proofs of our results.

Beck-type companion identities for Franklin's identity

“…It is straight forward to show that if (S 1 , S 2 ) is an Euler pair of order r, then | O j,r (n)| = | D j,r (n)|. In [7], we remark that a similar argument to [5] establishes analogues of Theorems 3 and 6 and thus analogues of Theorems 2 and 5 for all Euler pairs of order r. Denote by b j,r (n) the difference in the number of parts in O j,r (n) and the number of parts in D j,r (n). Denote by b ′ j,r (n) the analogous difference of the number of different parts.…”

“…Andrews [1] In [5], we showed that, if (S 1 , S 2 ) is an Euler pair, the identity of Theorem 7 has companion Beck-type identities analogous to (4) and (9).…”

“…The exponent of a part is the multiplicity of the part in the partition. For example, (5 2 , 4, 3 3 , 1 2 ) denotes the partition (5,5,4,3,3,3,1,1). Mostly, we will use the exponential notation when referring to rectangular partitions, i.e., partitions in which all parts are equal.…”

“…Since then, in a fairly short time, many articles appeared giving generalizations of this result as well as combinatorial proofs in many cases. See for example [11,18,4,12,13,14,3,5,6]. Some authors have started referring to these companion identities as Beck-type identities.…”