Almost partition identities

An almost partition identity is an identity for partition numbers that is true asymptotically100%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.

A cellular basis of the q-Brauer algebra related with Murphy bases of Hecke algebras

A new basis of the [Formula: see text]-Brauer algebra is introduced, which is a lift of Murphy bases of Hecke algebras of symmetric groups. This basis is shown to be a cellular basis in the sense of Graham and Lehrer. Using combinatorial tools we prove that the non-isomorphic simple [Formula: see text]-Brauer modules are indexed by the [Formula: see text]-restricted partitions of [Formula: see text] where [Formula: see text] is an integer, [Formula: see text]. When the [Formula: see text]-Brauer algebra has low dimension a criterion of semisimplicity is given, which is used to show that the [Formula: see text]-Brauer algebra is in general not isomorphic to the BMW-algebra.