A partition of a non-negative integer n is a way of writing n as a sum of a nondecreasing sequence of parts. The present paper provides the number of partitions of an integer n into parts of a specified number of different sizes. We establish new formulas for such partitions with particular interest to the number of partitions of n into parts of two sizes.… (More)

Let p(n) be the number of partitions of n. In this paper, we give a new identity for complete Bell polynomials based on a sequence related to the generating function of p(5n + 4) established by Srinivasa Ramanujan.

We study the posets (partially ordered sets) P, of partitions of an integer n, ordered by refinement, as defined by G. Birkhoff, “Lattice Theory” (3rd ed.) Colloq. Pub]. Vol. 25, 1967, Amer. Math. Sot. Providence. R.I. In particular we disprove the conjecture that the posets P,, are Cohen-Macaulay for all n. and show that even the Mobius function on the… (More)

Thomas E. Mason has shown that the number of ways in which a number n may be partitioned into consecutive parts, including the case of a single term, equals the number of odd divisors of n. This result is generalized by determining the number of partitions of n into arithmetic progressions with odd common difference, including the case of a single term.

Confidentiality was and will always remain a critical need in the exchanges either between persons or the official parties. Recently, cryptology has made a jump, from classical form to the quantum one, we talk about quantum cryptography. This theory, although is perfectly safe, there are still binding limits of implementation. In this paper, we developed a… (More)