A Combinatorial Interpretation of Punctured Partitions

We give a combinatorial interpretation of punctured partitions (i.e., n-tuples (p1, p2, …, pn) of natural numbers such that p1+p2+…+pk=k whenever pk≠0) in terms of linear partitions of linearly ordered sets. As an application we give an explicit expression of the permanent (determinant) of a particular kind of Hessenberg matrices in terms of punctured partitions (i.e., linear partitions). Then we show that for suitable choices of the Hessenberg matrix these permanents give the number of the enriched (linear) partitions of a finite (linearly ordered) set or more generally the associated polynomials forming a sequence of (Newjonian) binomial type. Instances of these polynomials are the exponential, rising factorial, Laguerre, Abel, inverse-Abel, Mittag–Leffler polynomials. A further application deals with formal series inversion; in particular we derive an expression of elementary symmetric functions in terms of complete symmetric functions and vice versa.