The Hankel Determinant of Exponential Polynomials

This paper appeared in American Mathematical Monthly 107 (2000), 557--560. Setting x = 1 in the identity gives the Hankel determinant of the Bell numbers. This case was mentioned by Martin Gardner in [1, p. 35]. Let R 0 , . . . , R n , C 0 , . . . , C n be disjoint sets such that the cardinality of R i and C i is i. We call R i the ith row set and C i the ith column set. Moreover, assume that the set R i possesses a linear order. Let S be the disjoint union of all the row and column sets. A partition # of the set S and a permutation # of the set {0, 1, . . . , n} are called compatible if for all blocks B in # there exists an index i such that B R i #(i) . Let T denote the set of all pairs (#, #), where # is a permutation and # is a compatible partition. Finally, let (-1) denote the sign of the permutation #. Proposition 2 The Hankel determinant of the exponential polynomials e n (x) has the following combinatorial interpretation: (#,#)#T x |#| . Proof: Consider the exponential polynomial e i+j (x) at row i and column j as a sum over all partitions of the set R i C j . Then the determinant can be expanded as # # 0 x |# # # #n x |#n | , where # i is a partition on the set R i #(i) . By letting # = # 0 # ## n , we obtain a partition on the set S that is compatible with the permutation #. Moreover, any pair (#, #) T may be obtained in this way. Finally, observe that |