Exchanges of "Quanta" Among Distinguishable and Indistinguishable Objects

Abstract

Beginning with a physical problem of exchange of n indistinguishable "quanta" of energy in an ensemble of k oscillators we define a new wide class of combinatorial problems, which also contains statistics intermediate between Fermi–Dirac and Bose–Einstein. One such problem is related to the number theoretic problem of computing the partitions of positive integers. After establishing such a connection, we give explicit formulas for the partitions M(n,k) of an integer n into k parts with k ≤ 4. Moreover, we derive a recursion relation for fixed n and varying k which is valid for any k. A formula which relates partitions to the cardinality of the partition set taking order into account is also derived. The leading asymptotic behavior for n large is obtained for any k. A suggestive interpretation of this formulas is proposed in terms of simplicial lattices. Recursive formulas at fixed k and varying n are then written for k ≤ 5 using the concept of factorial triangle, which is amenable for generalizations to larger k's. The problem of distinct partitions is mapped onto the probability problem of ball removal with replacement, for which we give again recursive solution formulas. Finally, the method of generalized Tartaglia triangle allows the derivation of recursive formulas for limited partitions which take order into account. This latter result is related to the problem of finding the number of distinct ways of dividing n indistinguishable objects into k distinguishable groups, for which explicit summations had been previously found.