D. Knuth used the
Robinson-Schensted “insertion into a tableau” algorithm to give a direct 1-to-1
correspondence between “generalized permutations” and ordered pairs of generalized
Young tableaux having the same shape. Since a generalized permutation
characterizes a power product of differential indeterminates, the work of D. Mead
on the principal differential ideal generated by a Wronskian provided an
independent proof of the existence of the Knuth bijection. This work led
Mead to suggest that other interesting combinatorial results may be found
by equating the cardinalities of different vector space bases for the same
finite-dimensional subspace of a differential ring. In a previous paper the
author showed how such combinatorial identities follow from the study of
“strong bases” for certain ideals in a ring of polynomials in a denumerable set
of indeterminates. The present paper completes that work by presenting
an infinite number of such strong bases and thus greatly expands the ring
theory and differential algebra having applications in the enumeration of
tableaux.
