Symmetric functions and P-recursiveness

Many enumeration problems, such as that of counting nonnegative integer matrices with given row and column sums, have solutions which can be expressed as coefficients of symmetric functions. We show here how useful formulas can be obtained from these symmetric function generating functions. In some cases, the symmetric functions yield reasonably simple explicit formulas or generating functions for the coefficients. In other cases, such as in counting several types of regular graphs, explicit formulas that are too unwieldy to be useful in computation can still be used to show that a sequence of coefficients is P-recursive, that is, it satisfies a linear homogeneous recurrence with polynomial coefficients. In Sections 2 and 3 we review the basic facts about symmetric functions and introduce the method we use (due to Read [35]) for coefficient extraction: The coefficient of x1 i1 x λ2 i2 · · ·xk ik in the symmetric function f is the scalar product 〈f, hλ1 · · ·hλk〉, where hn is the complete symmetric function. To evaluate this scalar product, we expand f and the hλi in power sum symmetric functions, and use the orthogonality of the power sum symmetric functions. We give examples of explicit formulas and generating functions obtained by this method for problems of counting permutations, trees, and partitions. In Sections 4 and 5 we introduce the basic facts about P-recursive functions and D-finite power series. We define D-finite symmetric functions, and show that their coefficients give rise to P-recursive sequences, proving a conjecture of Gouden and Jackson [18] that the counting sequence for k-regular graphs is P-recursive for all k. In Section 6, we apply the theory to Schur functions, and use some formulas of Gordon and Houten [15], Gordon [16], and Bender and Knuth [5] to show how the exponential generating function for the number of standard tableaux with at most k rows can be expressed in terms of a determinant of Bessel functions. We obtain the explicit formulas of Regev [39] and