Combinatorial Applications of Hermite Polynomials

Let $C_1 ,C_2 , \cdots ,C_k $ be k finite sets of elements, where $n_i $ is the number of elements in $C_i (i = 1,2, \cdots ,k)$ and $\sum_{i = 1}^k {n_i } $ is even, $2S$ (say). In any arrangement of the elements into S disjoint pairs, we count the number of homogeneous pairs, i.e., those in which both numbers are from the same subset, $C_i $. We define such a pairing as even, odd or pure according as the number of homogeneous pairs is even, odd or zero respectively. The numbers of possible pairings of the different types are expressed as integrals involving Hermite polynomials, and these expressions are used both to derive new combinatorial results and also to provide combinatorial proofs of analytical formulae.