The posets I(n) and M(n): Probabilities of binary strings and partitions into distinct parts

The posetM(n) of all partitions of integers into distinct parts less than or equal to n has been widely studied during the last three decades. Among other properties, M(n) is self-dual, graded, rank-symmetric, rank-unimodal, strongly Sperner and, therefore, Peck [3, 4, 5]. This work presents a new description of M(n) through its isomorphic poset I(n) := ({0, 1} , ), that we have introduced in the context of the modelling of stochastic Boolean systems [1]. The so-called intrinsic order, “ ”, extends to lexicographic order, respects the Hamming weight, and sorts the 2 binary n-tuples (associated to the n-dimensional Bernoulli distribution) by their occurrence probabilities just looking at the positions of their 0s and 1s. Two previous results are required: Simple matrix characterizations of both, the intrinsic order and the covering relation in I(n) [1, 2]. We prove that I(n) is graded of rank Tn (n-th triangular number), with rank function