Generalizations of the Cauchy and Schur identities

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Abstract

One of the generalizations proved is that pk(π)z(π)=p+n−1p−1n−1p−1 where the summation is over all partitions π = (1k1,…, nkn) of n with parts not divisible by p, k(π) = k1 + … + kn and z(π) = 1k1k1! … nknkn!. p = 1 gives the Cauchy identity and p = 2 the Schur identity. This identity is itself obtained as a particular case of a more general identity and the proof involves a generalization of certain symmetric functions, called Hall functions, which have played a major role in recent enumeration problems.

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