11 papers
Strong -log-convexity of the -Hoggatt polynomials
Shane Chern, Wenle Shi
In this paper, three (-)log-convexity or concavity properties related to the -Hoggatt numbers are investigated. First, we show that the -Hoggatt transformation preserves t…
On the occurrence of congruence multiplicities between Ramanujan's theta functions
Shane Chern, Nicolas Allen Smoot, Dazhao Tang
Recently, the first and third authors initiated a study of arithmetical relationships between Ramanujan's theta functions and . In this work, we prove eight families…
Hankel determinants of Catalan-like sequences
Shane Chern, Wenle Shi
In this paper, we compute the (shifted) Hankel determinants of Catalan-like sequences, which arise naturally from the weighted enumerations of nonintersecting Motzkin meanders. Amo…
Convolutive sequences, II: Parametrizations
Shane Chern, Dennis Eichhorn, Shishuo Fu +1
In recent work, the authors defined a sequence to be -convolutive exactly if \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end…
Tadpole Nahm sum as a Wronskian
Shane Chern, Chanh Tran, Tanay Wakhare
We show that the tadpole Nahm sum is essentially a specialization of a series studied by Bartlett and Warnaar. This overlooked connection was first discovered by the internal model…
Proof of Cigler's conjecture on -Hoggatt numbers
Shane Chern, Wenle Shi
We prove the nonnegativity and palindromicity of a family of polynomials arising from -Hoggatt numbers. The nonnegativity is derived from Stanley's -partition theory thro…
Finite Kleshchev bipartitions and -trinomial coefficients
Shane Chern
The Kleshchev multipartitions arise in the representation theory for the Ariki-Koike algebras. In previous work, Li, Stanton, Xue, Yee, and the author considered a refined enumerat…
On conjectural fermionic formulas for the Macdonald index in Argyres-Douglas theories
Shane Chern, Chanh Tran, Tanay Wakhare
We prove a fermionic-bosonic duality relation for the Macdonald index in Argyres-Douglas theories of type , thereby yielding a conjectural fermionic formula due to…
Signed counting of partition matrices
Shane Chern, Shishuo Fu
We prove that the signed counting (with respect to the parity of the ``'' statistic) of partition matrices equals the cardinality of a subclass of inversion seq…
Convolutive sequences, I: Through the lens of integer partition functions
Shane Chern, Dennis Eichhorn, Shishuo Fu +1
Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}…
Ramanujan's theta functions and internal congruences modulo arbitrary powers of
Shane Chern, Dazhao Tang
In this work, we investigate internal congruences modulo arbitrary powers of for two functions arising from Ramanujan's classical theta functions and . By letting…