Matricial Formulae for Partitions

X iv :0 80 6. 12 73 v1 [ m at h. N T ] 7 J un 2 00 8 MATRICIAL FORMULAE FOR PARTITIONS F. AICARDI Abstract. The exponential of the triangular matrix whose entries in the diagonal at distance n from the principal diagonal are all equal to the sum of the inverse of the divisors of n is the triangular matrix whose entries in the diagonal at distance n from the principal diagonal are all equal to the number of partitions of n. A similar result is true for any pair of sequences satisfying a special recurrence. Let ∆ be the infinite triangular matrix having zeroes on and below the main diagonal, and whose values on each parallel to the main diagonal are all equal to: ∆i,i+n = σ(n) (i ≥ 0, n > 0), where σ(n) is the sum of the inverses of the divisors of n, for n > 0. Proposition 1. The matrix P = exp(∆) is triangular, with zero below the diagonal, 1 on the diagonal, and its elements on the parallel to the diagonal at distance n from the diagonal are equal to the number p(n) of partitions of the integer n (being p(0) = 1): Pi,i+n = p(n) (i ≥ 0, n ≥ 0).