Partitions of large unbalanced bipartites

Abstract

Abstract. We compute the asymptotic behaviour of the number of partitions of large vectors (n1, n2) of Z 2 + in the critical regime n1 √ n2 and in the subcritical regime. This work completes the results established in the fifties by Auluck, Nanda, and Wright.

Extracting Aggregation Free Energies of Mixed Clusters from Simulations of Small Systems: Application to Ionic Surfactant Micelles

“…The possible values of m j , k such that correspond to the set of partitions of the bipartite number ( N A , N B ). The number of bipartite partitions as a function of N A and N B has been studied , and found to be significantly larger than the number of regular integer partitions of N A + N B . While algorithms to generate integer partitions have been described, to our knowledge no such algorithm for bipartite partitions has been published; an algorithm that we developed for this purpose is presented in the Supporting Information.…”

“…Results from several trajectories with limited numbers of particles were combined to give a globally consistent freeenergy surface describing the growth of methyl t-butyl ether (MTBE) clusters along the pathway to nucleation of a bulk liquid phase from either the vapor or aqueous solution. The analysis of equilibrium cluster statistics in small-N systems relies on the consideration of all discrete ways in which the monomers of the system can be arranged into clusters, 26 which can be mapped onto the integer partitions of N. The number of these partitions scales as (with a, b, and c being explicit constants) for partitions of bipartite integers, 28,29 which represent the combinations of clusters that can be formed from a two-component mixture of aggregants. Explicit generation of these partitions, in the latter case in particular, becomes computationally impractical even for modest N. A new approach, derived using techniques commonly used in number theory, enables the evaluation of the appropriate sums over sets of partitions without explicitly generating those partitions, resulting in a more efficient algorithm whose computational cost scales better than N 3 for single component systems and better than N 6 for bipartite mixtures.…”

“…The analysis of equilibrium cluster statistics in small- N systems relies on the consideration of all discrete ways in which the monomers of the system can be arranged into clusters, which can be mapped onto the integer partitions of N . The number of these partitions scales as for simple integers and as (with a , b , and c being explicit constants) for partitions of bipartite integers, , which represent the combinations of clusters that can be formed from a two-component mixture of aggregants. Explicit generation of these partitions, in the latter case in particular, becomes computationally impractical even for modest N .…”

$$ \mathcal{N} $$ = 2* Schur indices

“…then the coefficient p(n, N ) is the number of partitions of n with N parts [72]. It implies the correspondence between the 1/2 BPS local operators of dimension n in N = 4 U(N ) SYM theory and partitions of n with N parts.…”

On the number of lattice convex chains

“…Under this assumption, the oscillating term I crit ((κ/n) 1/3 ) is at most of order O(n 1 6 + ) for all > 0. Indeed, the right-side γ right of the contour of integration in formula (7) can be replaced by a vertical line arbitrarily close to the critical line. This writing implies that, for all > 0, I crit (β) = O(β − 1 2 − ) as β goes to 0.…”