We find a close correspondence between the partition functions of ideal quantum gases and certain symmetric polynomials. From this correspondence, it can be shown that a number of thermodynamic identities that have recently been considered in the literature are essentially of combinatorial origin and have been known for a long time as theorems on symmetric polynomials. For example, a recurrence relation for partition functions in the textbook by P. Landsberg is Newton’s identity in disguised form. Conversely, a theorem on symmetric polynomials translates into a new and unexpected relation between fermion and boson partition functions, which can be used to express the former by means of the latter and vice versa.
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