3 research outputs found
Differential operators, grammars and Young tableaux
In algebraic combinatorics and formal calculation, context-free grammar is
defined by a formal derivative based on a set of substitution rules. In this
paper, we investigate this issue from three related viewpoints. Firstly, we
introduce a differential operator method. As one of the applications, we deduce
a new grammar for the Narayana polynomials. Secondly, we investigate the normal
ordered grammars associated with the Eulerian polynomials. Thirdly, motivated
by the theory of differential posets, we introduce a box sorting algorithm
which leads to a bijection between the terms in the expansion of and
a kind of ordered weak set partitions, where is a smooth function in the
indeterminate and is the derivative with respect to . Using a map
from ordered weak set partitions to standard Young tableaux, we find an
expansion of in terms of standard Young tableaux. Combining this with
the theory of context-free grammars, we provide a unified interpretations for
the Ramanujan polynomials, Andr\'e polynomials, left peak polynomials, interior
peak polynomials, Eulerian polynomials of types and , -Eulerian
polynomials, second-order Eulerian polynomials, and Narayana polynomials of
types and in terms of standard Young tableaux. Along the same lines, we
present an expansion of the powers of in terms of standard Young
tableaux, where is a positive integer. In particular, we provide four
interpretations for the second-order Eulerian polynomials. All of the above
apply to the theory of formal differential operator rings.Comment: 38 page