A context-free grammar for the Ramanujan-Shor polynomials

Abstract

Ramanujan defined the polynomials ψ k (r, x) in his study of power series inversion. Berndt, Evans and Wilson obtained a recurrence relation for ψ k (r, x). In a different context, Shor introduced the polynomials Q(i, j, k) related to improper edges of a rooted tree, leading to a refinement of Cayley's formula. He also proved a recurrence relation and raised the question of finding a combinatorial proof. Zeng realized that the polynomials of Ramanujan coincide with the polynomials of Shor, and that the recurrence relation of Shor coincides with the recurrence relation of Berndt, Evans and Wilson. So we call these polynomials the Ramanujan-Shor polynomials, and call the recurrence relation the Berndt-Evans-Wilson-Shor recursion. A combinatorial proof of this recursion was obtained by Chen and Guo, and a simpler proof was recently given by Guo. From another perspective, Dumont and Ramamonjisoa found a context-free grammar G to generate the number of rooted trees on n vertices with k improper edges. Based on the grammar G, we find a grammar H for the Ramanujan-Shor polynomials. This leads to a formal calculus for the Ramanujan-Shor polynomials. In particular, we obtain a grammatical derivation of the Berndt-Evans-Wilson-Shor recursion. We also provide a grammatical approach to the Abel identities and a grammatical explanation of the Lacasse identity.

Total positivity of some polynomial matrices that enumerate labeled trees and forests, I. Forests of rooted labeled trees

“…(Of course, it was redundant to introduce the two variables y and z instead of just one of them; we did it because it makes the formulae more symmetric.) In particular, the polynomials f n,1 (y, z) enumerate rooted trees according to the number of improper edges; they are homogenized versions of the celebrated Ramanujan polynomials [21,37,62,63,69,80,98,111,128] [91, A054589]. 8 The unit-lower-triangular matrix (f n,k (y, z)) n,k≥0 is also the exponential Riordan array R[F, G] with F (t) = 1 and…”

Trees, Forests, and Total Positivity: I. $q$-Trees and $q$-Forests Matrices

“…Note that the weight function (3.11) defined above is for forests on the vertex set [n] with k components where k < n. For k = n the set F n,n consists of a single forest on the vertex set [n] consisting of n components, to which we assign a weight of 1. 9 The weight of a forest F ∈ F n,k for general n and k is therefore defined to be wt(F ) := 1 if k = n, r niblings(F ) s scindex(F ) t parents(F )−(n−k−1) if k < n.…”

“…each entry is a homogeneous polynomial of degree n − k, and under the specialisation y = z = 1 we recover the forests matrix (that is, F(1, 1) = F). Sokal also observes (see the remark on page 7 of [66]) that the polynomials f n,1 (y, z) (which enumerate rooted trees with respect to improper edges) are homogenised versions of the Ramanujan polynomials (see [9,20,35,36,40,45,55,62,75] and [63, A054589]). The matrix F(y, z) is the exponential Riordan array R[F, G ] with F (t) = 1 and [66]) where G(t) is the tree function (see [16] and (1.11) above).…”

Trees, forests, and total positivity: I. $q$-trees and $q$-forests matrices

“…We will briefly touch upon the method of production matrices (see [18,19]), which in recent years has become an important tool in enumerative combinatorics and has its roots in Stieltjes' work on continued fractions 9 . The theory of production matrices with respect to total positivity is extensively studied in [64], however, since [64] is not yet publicly available we direct the reader to Sections 2.2 and 2.3 of [66] for a fuller treatment.…”

“…each entry is a homogeneous polynomial of degree n − k, and under the specialisation y = z = 1 we recover the forests matrix (that is, F(1, 1) = F). Sokal also observes (see the remark on page 7 of [66]) that the polynomials f n,1 (y, z) (which enumerate rooted trees with respect to improper edges) are homogenised versions of the Ramanujan polynomials (see [9,20,35,36,40,45,55,62,75] and [63, A054589]). The matrix F(y, z) is the exponential Riordan array R[F, G ] with F (t) = 1 and…”