2 research outputs found
A combinatorial conjecture from PAC-Bayesian machine learning
We present a proof of a combinatorial conjecture from the second author's
Ph.D. thesis. The proof relies on binomial and multinomial sums identities. We
also discuss the relevance of the conjecture in the context of PAC-Bayesian
machine learning.Comment: 6 page
A Context-free Grammar for the Ramanujan-Shor Polynomials
Ramanujan defined the polynomials in his study of power
series inversion. Berndt, Evans and Wilson obtained a recurrence relation for
. In a different context, Shor introduced the polynomials
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 to generate the number of rooted trees on
vertices with improper edges. Based on the grammar , we find a grammar
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.Comment: 21 pages, 4 figure