EXCHANGES OF QUANTA AMONG DISTINGUISHABLE AND INDISTINGUISHABLE OBJECTS

Beginning with a physical problem of exchange of n indistinguishable ' 'quanta'' of energy in an ensemble of k oscillators we define a new wi de class of combinatorial problems, which also contains statistics int ermediate between Fermi-Dirac and Bose-Einstein. One such problem is r elated to the number theoretic problem of computing the partitions of positive integers. After establishing such a connection, we give expli cit formulas for the partitions M(n, k) of an integer n into k parts w ith k less than or equal to 4. Moreover, we derive a recursion relatio n 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 orde r into account is also derived. The leading asymptotic behavior for n large is obtained for any ic. A suggestive interpretation of this form ulas is proposed in terms of simplicial lattices. Recursive formulas a t fixed k and varying n are then written for k less than or equal to 5 using the concept of factorial triangle, which is amenable for genera lizations to larger k's. The problem of distinct partitions is mapped onto the probability problem of ball removal with replacement, for whi ch we give again recursive solution formulas. Finally, the method of g eneralized Tartaglia triangle allows the derivation of recursive formu las for limited partitions which take order into account. This latter result is related to the problem of finding the number of distinct way s of dividing n indistinguishable objects into k distinguishable group s, for which explicit summations had been previously found.