arXiv subjects
Publications and source records attributed to Shane Chern.
Using a rational analogue of an $\mathrm{A}_2^{(1)}$ Rogers--Ramanujan type identity discovered by Warnaar, we establish sum-like expressions for a bivariate generating function for cylindric partitions of rank three. These expressions provide the finishing touch towards conjectures of Warnaar and Kanade--Russell.
In this paper, three ($q$-)log-convexity or concavity properties related to the $d$-Hoggatt numbers are investigated. First, we show that the $d$-Hoggatt transformation preserves the log-convexity of a sequence. We then establish the strong $q$-log-concavity of a $q$-analog of $d$-Hoggatt numbers. Finally, we prove the core result of this work, namely, the sequence of $d$-Hoggatt polynomials is strongly $q$-log-convex, based on the theory of Schur functions.
Recently, the first and third authors initiated a study of arithmetical relationships between Ramanujan's theta functions $φ(-q)$ and $ψ(q)$. In this work, we prove eight families of internal congruences modulo arbitrary powers of $3$ and $5$ for infinite series related to the two theta functions. We also show that there exist isomorphisms between the congruence families for $φ(-q)$ and $ψ(q)$, which can be realized by the study of congruence multiplicities previously studied by Garvan, Sellers, and the second author. We believe that such equivalences are exclusive, at least on the congruence subgroups $Γ_0(6)$ and $Γ_0(10)$. In the end, we show how a simple manipulation of function field extensions allows us to predict whether additional isomorphisms to our congruences occur.
In this paper, we compute the (shifted) Hankel determinants of Catalan-like sequences, which arise naturally from the weighted enumerations of nonintersecting Motzkin meanders. Among these determinant evaluations, one and a half are newly discovered, featuring generic shifted Hankel determinants; two were formulated earlier by Cigler and Krattenthaler in an equivalent combinatorial form; and the rest were conjectured by Cigler. As an application, we further confirm a conjectural binomial determinant identity proposed by Cigler and Krattenthaler.
We establish several new series evaluations involving the central $q$-binomial coefficients, with the inspiration coming from earlier work by Vignat and one of the authors on the limiting case at $q\to 1$.
In recent work, the authors defined a sequence $(a_n)_{n\ge 0}$ to be $m$-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{align*} for a specific positive integer $m$ and provided proofs of the $2$- and $3$-convolutivity of a small number of sequences arising from primitive eta-products. Since the completion of that work, the authors have discovered many new instances of convolutive eta-products. The main focus of this work is to unify all but one of these instances in a parametric way.
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 mathvision-harness-0.3 at MathVision AI. We then use Macdonald-type identities for the Bartlett-Warnaar series to express the principal tadpole Nahm sum and its twisted version in terms of a Wronskian of generalized theta series. The principal case confirms a conjecture of Milas and Wang.
Russell recently introduced a family of three-colored partitions in analogy with earlier work by Alladi and Gordon. He showed that their enumerations are surprisingly connected to the classic Rogers-Ramanujan identities. Using the method of linked partition ideals, we elaborate on Russell's results by further counting each part color. Unlike Russell's proofs, our arguments do not require any use of computer algebra systems. With these multivariate generating function relations, we are led to confirm a conjecture of Russell. Furthermore, our results offer a combinatorial understanding of a new triple Rogers-Ramanujan type identity.
We prove the nonnegativity and palindromicity of a family of polynomials arising from $q$-Hoggatt numbers. The nonnegativity is derived from Stanley's $(P,ω)$-partition theory through a standard Young tableau formula, while the palindromicity is proved by an involution on rectangular standard Young tableaux. Our result confirms a conjecture of Cigler.
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 enumeration for the $2$-dimensional case, namely, the Kleshchev bipartitions, by invoking the $2$-residue statistic for partitions. In this paper, we make further elaboration by bounding the largest part of the bipartitions and show that the related counting functions are connected with two families of $q$-trinomial coefficients introduced by Andrews and Baxter.
We prove a fermionic-bosonic duality relation for the Macdonald index in Argyres-Douglas theories of type $(A_1, D_{2k+1})$, thereby yielding a conjectural fermionic formula due to Andrews et al. Our duality is built upon a new conjugate Bailey pair to be established using techniques from orthogonal polynomials and basic hypergeometric series. In addition, this fermionic formula implies another sum-like expression independently conjectured by Andrews et al. and Kim et al. for the same Macdonald index.
We prove that the signed counting (with respect to the parity of the ``$\operatorname{inv}$'' statistic) of partition matrices equals the cardinality of a subclass of inversion sequences. In the course of establishing this result, we introduce an interesting class of partition matrices called improper partition matrices. We further show that a subset of improper partition matrices is equinumerous with the set of Motzkin paths. Such an equidistribution is established both analytically and bijectively.
We prove determinantal-Pfaffian formulae that simultaneously generalise the Pfaffian minor summation formula of Ishikawa and Wakayama and Byun's recent minor summation formula. These formulae are based on factorisation formulae for the determinant of the sum of a skew-symmetric matrix and a rank-1 matrix. Applications include a Cauchy-type identity for skew Schur functions.
We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.
We prove that for $k\ge 1$, all coefficients in the expansion of the series $$\sum_{n\ge 0} \frac{(q^{2n+2}, q^{2n+2k}; q^2)_\infty}{(q^{2n+1};q^2)_\infty^2} q^{2n}$$ are positive, by $q$-hypergeometric means. This confirms a recent conjecture of Andrews and El Bachraoui.
Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}$ of primitive eta-products that satisfy the generic convolutive property \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$. Given the results of an exhaustive search of the Online Encyclopedia of Integer Sequences for such sequences for $m$ up to $6$, we first focus on the case where $m=2$ with our attention mainly paid to the combinatorics of two $2$-convolutive sequences, featuring bijective proofs for both. For other $2$-convolutive sequences discovered in the OEIS, we apply generating function manipulations to show their convolutivity. We also give two examples of $3$-convolutive sequences. Finally, we discuss other convolutive series that are not eta-products.
We compute the Quot and finitized Coh zeta functions of the inert quadratic orders $\mathbb{F}_q[[T]]+T^{m}\mathbb{F}_{q^{2}}[[T]]$ for every $m\geq 1$ in terms of a $2m$-fold multisum, and then show this multisum equals an $m$-fold Bressoud sum. This proves a recent conjecture of the second author, rounding up the line of exploration in the series of work by the authors and Jiang. The equality between the $2m$-fold multisum and the $m$-fold Bressoud sum is built upon generalizing the multisum by introducing a ``ghost'' parameter $a$ to its summands. We then show that such an $a$-generalization is surprisingly $a$-independent by purely $q$-theoretic techniques. Finally, we propose a refined multisum that interpolates two versions of Quot zeta functions for all three types of quadratic orders.
Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determinants in terms of domino tilings and nonintersecting lattice paths, thereby partially answering a question of theirs. Furthermore, the determinant evaluations established by Koutschan, Krattenthaler and Schlosser produce many product formulas for our weighted enumerations of domino tilings and nonintersecting lattice paths. However, there are still two enumerations left corresponding to conjectural formulas made by the three. We hereby prove the two conjectures using the principle of holonomic Ansatz plus the approach of modular reduction for creative telescoping, and hence fill the gap.