Decomposition of Triply Rooted Trees

Chen, William Y. C.; Peng, Janet F. F.; Yang, Harold R. L.

doi:10.37236/3016

Cited by 8 publications

(5 citation statements)

References 9 publications

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“…Notice that the label S indicates where one can apply this operation. Case 2: T 2 is obtained from T by splitting the left edge (4, 1) into two edges (4, 5) and (5,1). This operation corresponds to the substitution rule A → A 3 S.…”

Section: The Dumont-ramamonjisoa Grammarmentioning

confidence: 99%

“…Since then, several proofs have been found. For example, Sun [18] gave a derivation by using the umbral calculus, Younsi [19] found a proof with aid of the Abel identity, Prodinger [13] provided a justification based on Cauchy's integral formula, Gessel [8] proved the identity by means of the Lagrange inversion formula, and Chen, Peng and Yang [5] obtained a combinatorial interpretation in terms of triply rooted trees. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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A context-free grammar for the Ramanujan-Shor polynomials

Chen

Yang

2021

Advances in Applied Mathematics

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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.

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Section: The Dumont-ramamonjisoa Grammarmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A context-free grammar for the Ramanujan-Shor polynomials

Chen

Yang

2021

Advances in Applied Mathematics

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“…[2], [5], [20], [23]), the proof we present remains, as far as we know, the only one providing equivalent expressions for the functions ξ and ξ 2 that are simpler and more convenient from a numerical perspective. Furthermore, none of the aforementioned papers [2], [5], [20], [23] discuss the context and relevance of Conjecture 2.1 in the theory of machine learning.…”

mentioning

confidence: 90%

A combinatorial conjecture from PAC-Bayesian machine learning

Younsi,

Lacasse

2020

Preprint

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“…Recently, by applying the Hurwitz identity on multivariate Abel polynomials, Younsi [7] gave an algebraic proof of this conjecture. Later, using a decomposition of triply rooted trees into three doubly rooted trees, Chen, Peng and Yang [1] gave it a nice combinatorial interpretation.…”

Section: Introductionmentioning

confidence: 99%