The Janjić–Petković Inset Counting Function: Riordan Array Properties and a Thermodynamic Application

Let $q_1+\dots +q_n+m$ objects be arranged in n rows with $q_1,\dots ,q_n$ objects and one last row with m objects. The Janjić–Petković counting function denotes the number of $(n+k)$-insets, defined as subsets containing $n+k$ objects such that at least one object is chosen from each of the first n rows, generalizing the binomial coefficient that is recovered for $q_1$=…=$q_n$= 1, as then only the last row matters. Here, we discuss two explicit forms, combinatorial interpretations, recursion relations, an integral representation, generating functions, convolutions, special cases, and inverse pairs of summation formulas. Based on one of the generating functions, we show that the Janjić–Petković counting function, like the binomial coefficients that it generalizes, may be regarded as a Riordan array, leading to additional identities. As an application to a physical system, we calculate the heat capacity of a many-body system for which the configurations are constrained as described by the Janjić–Petković counting function, resulting in a modified Schottky anomaly.