Multisectioned partitions of integers

Abstract

This paper presents identities on generating functions for multisectioned partitions of integers by developing in the language of partitions some powerful and essentially combinatorial techniques from the literature of principal differential ideals. D. Mead has stated in Vol. 42 of this journal that one can obtain interesting combinatorial relations by constructing different vector space bases for a subspace of a differential ring and using the fact that the cardinality of all bases is the same. The results of the present paper are of this nature. In particular, we enumerate certain sets of ordered pairs of that have a central role in Mead's paper. Tableaux were used by A. Young and others to study the structure of the symmetric groups Sn. In [3], D. Knuth used an insertion into tableau construction of C. Schensted to give a direct 1-to-l correspondence between generalized permutations and ordered pairs of generalized Young tableaux having the same shape. In [5], Mead independently proved the existence of such a bisection while developing a new vector space basis for the ring of differential polynomials in n independent differential indeterminates. Mead's paper deals with principal differential ideals generated by Wronskians and used determinantal identities going back to Gayley. The ordered pairs of used by Mead appear in a more general setting in the paper [1] by Doubilet, Rota, and Stein.