For integers $r\ge2$ and $n\ge1$, let $$ F_{r,n}(q)=\prod_{k=1}^{n}\frac{1-q^{rk}}{1-q^k}. $$ The coefficient of $q^j$ in \(F_{r,n}(q)\) counts partitions of $j$ into parts at most $n$, each occurring at most $r-1$ times. Hughes proved that $F_{2,n}(q)=\prod_{k=1}^{n}(1+q^k)$ is unimodal for every $n\ge 1$. This result was reproved by Stanley using an algebraic approach and Odlyzko and Richmond using an analytic approach. Almkvist conjectured that $F_{r,n}(q)$ is unimodal in the following two cases: every even $r$ and every $n\ge 1$; every odd $r$ and every $n\ge 11$. He proved that this conjecture is true for $3\le r \le 20$ and $r=100,101$. In this paper, we completely settle the conjecture.
Let $c\phi_k(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. We prove that, for every $m\geq2$ and every $k\equiv2\pmod{2^m}$, \[ \sum_{n\geq0}c\phi_k(n)q^n\equiv\frac{\varphi(q)\,(q^2;q^2)_\infty}{(q;q)_\infty^2}\sum_{n\geq0}c\phi_{k/2}(n)q^{2n}\pmod{2^m}, \] where $\varphi(q)$ is the cla...
Let $R$ be a commutative ring and $f(a_1, \ldots, a_n) = \sum_{i=1}^k r_{i1}^{a_1} \cdots r_{in}^{a_n} m_i$ be a linear-exponential map over an $R$-module $M$. Dong and Shafrir (2026) showed that, when $\ell M = 0$ for some $\ell \in \mathbb{N}_{>0}$, the zero set of $f$ is the intersection of effectively computable $p...
Let $W=W(B_n)$ act diagonally on $\mathfrak{h}\oplus\mathfrak{h}^*$, let $S=\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^*]$, let $J\subset S$ be the ideal generated by the $W$-alternating polynomials and $\mathfrak{m}_S$ is the maximal ideal of the origin. For sufficiently large $m$ we compute $q,t$-Fuss-Catalan polynomi...
For every negative fundamental discriminant $-f$ and every $k\geq1$, we construct a rational function $R_{f,2k}\in\mathbb{Q}(x_1,\ldots,x_{2k})$ and a constant $r_{f,2k}\in\mathbb{Q}^\times$ such that \[ m(R_{f,2k})=r_{f,2k}L'(\chi_{-f},1-2k), \] where $m$ denotes the logarithmic Mahler measure and $\chi_{-f}$ is the q...
Let $(a_n)_{n\geqslant 0}$ be a sequence of integers. Its dual sequence $(a_n^*)_{n\geqslant 0}$ is defined by \begin{equation*} a_n^* := \sum_{k=0}^{n} \binom{n}{k}(-1)^k a_k. \end{equation*} Let $p>3$ be a prime. In this paper we mainly investigate congruences modulo $p^2$ involving central binomial coefficients and...
For an integer $k \ge 0$, let $f_k(x) = 1 + e(x) + e((k+2)x)$ and $g_k(x) = 1 + e(x) - e((k+2)x)$ on $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, where $e(x) = e^{2\pi i x}$. Mockenhaupt conjectured that $\|g_k\|_{L^p(\mathbb{T})}>\|f_k\|_{L^p(\mathbb{T})}$ whenever $2k<p<2k+2$. The conjecture was previously known for $k \le...
Guan-Cheng Pan, Cheng-Song You, Heng-Yu Wang et al.· 0 citations
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