Grammaire de Ramanujan et Arbres de Cayley

Dumont, Dominique; Ramamonjisoa, Armand

doi:10.37236/1275

Cited by 33 publications

(48 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Dumont and Ramamonjisoa [12] independently found the same combinatorial interpretation for the coefficients of the Ramanujan polynomial R n (x). Let r ′ (n, k) = r(n+1, k).…”

Section: The Ramanujan Polynomialsmentioning

confidence: 70%

Recurrence Relations for Strongly q-Log-Convex Polynomials

2011

View full text Add to dashboard Cite

show abstract

“…Dumont and Ramamonjisoa [12] independently found the same combinatorial interpretation for the coefficients of the Ramanujan polynomial R n (x). Let r ′ (n, k) = r(n+1, k).…”

Section: The Ramanujan Polynomialsmentioning

confidence: 70%

Recurrence Relations for Strongly q-Log-Convex Polynomials

2011

View full text Add to dashboard Cite

show abstract

“…In the notation of Q n (x, y), the relation (2.3) of Dumont and Ramamonjisoa takes the form D n−1 (yw) = y n w n Q n (0, y). (5.4) In addition, Dumont [7] obtained grammatical expressions of Q n (1, y) and Q n (−1, y):…”

Section: The Abel Identitiesmentioning

confidence: 99%

A context-free grammar for the Ramanujan-Shor polynomials

Chen

Yang

2021

Advances in Applied Mathematics

View full text Add to dashboard Cite

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.

show abstract

“…However, Foata and Han [11] later found a way to compute the generating function of E n (x, 1) without solving a differential equation. Dumont [9] also discovered the following grammar for the Ramanujan polynomials:…”

Section: A Grammatical Labeling Of Permutationsmentioning

confidence: 99%

A context-free grammar for peaks and double descents of permutations

2018

Advances in Applied Mathematics

View full text Add to dashboard Cite

This paper is concerned with the joint distribution of the number of exterior peaks and the number of proper double descents over permutations on [n] = {1, 2, . . . , n}. The notion of exterior peaks of a permutation was introduced by Aguiar, Bergeron and Nyman in their study of the peak algebra. Gessel obtained the generating function of the number of permutations on [n] with a given number of exterior peaks. On the other hand, by establishing differential equations, Elizalde and Noy derived the generating function for the number of permutations on [n] with a given number of proper double descents. Barry and Basset independently deduced the generating function of the number of permutations on [n] with no proper double descents. We find a context-free grammar which can be used to compute the number of permutations on [n] with a given number of exterior peaks and a given number of proper double descents. Based on the grammar, the recurrence relation of the number of permutations on [n] with a give number of exterior peaks can be easily obtained. Moreover, we use the grammatical calculus to derive the generating function without solving differential equations. Our formula reduces to the formulas of Gessel, Elizalde-Noy, Barry, and Basset. Finally, from the grammar we establish a relationship between our generating function and the generating function of the joint distribution of the number of peaks and the number of double descent derived by Carlitz and Scoville.

show abstract

Grammaire de Ramanujan et Arbres de Cayley

Cited by 33 publications

References 0 publications

Recurrence Relations for Strongly q-Log-Convex Polynomials

Recurrence Relations for Strongly q-Log-Convex Polynomials

A context-free grammar for the Ramanujan-Shor polynomials

A context-free grammar for peaks and double descents of permutations

Contact Info

Product

Resources

About