Recently, Andrews and El Bachraoui studied a partition function $s_1(n)$, which counts the number of two-color partitions into distinct parts of $n$ whose smallest part occurs in one prescribed color only, while every larger part may occur in either color or in both colors. They obtained a complete description modulo 4 for $s_1(n)$. They also considered a $q$-series $T_{o}(q)$ which is the odd companion series of the generating function for $s_1(n)$. At the end of their paper, they presented a conjecture on the parity of the coefficients of $T_o(q)$. In this paper, we confirm this conjecture. Moreover, we establish an infinite family of congruences modulo 8 for the coefficients of $S_1(q)$ and prove that the set of integers satisfying $s_1(n)\equiv 0\pmod 8$ has natural density one.
Andrews and Bachraoui (\textit{Int. J. Number Theory} (2026)) studied the two-color partition function $F(n)$ of a non-negative integer $n$ wherein odd parts may appear in two colors (red and blue) and even parts appear in one color (blue). For any non-negative integer $n,$ they also considered some restricted versions...
In this paper, we prove a conjecture of Andrews and El Bachraoui concerning the parity of certain two-color partitions. Precisely, we show that if the Fourier coefficient $t_o(n)$ of the corresponding $q$-series is odd, then $8n+9$ is represented by the binary quadratic form $x^2+2y^2$.
In this paper, we prove an exceptional sign pair phenomenon for three $q$-series from Ramanujan's Lost Notebook, namely $v_2(q),v_3(q)$, and $v_4(q)$. This was first observed by Andrews in a 1986 paper. Building on the recent work of Kundu, Storzer, Wang, and the author, which established that the coefficients of these...
Let $a(n)$ be the sequence A028342 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function $\sum_{n\ge0} a(n)x^n/n! = \prod_{i\ge1}(1-x^i)^{-1/i}$. Equivalently, $a(n)$ counts permutations of an $n$-element labeled set in which every cycle is assigned one divisor of its l...
Let $K/k$ be a finite abelian extension of number fields with Galois group $G$ and let $n\geq 2$. We prove, assuming the relevant $p$-part of the equivariant Tamagawa number conjecture, first-derivative analogues of the Deligne-Ribet integrality theorem and of the Coates-Sinnott conjecture. We construct a rank-one lead...
Stone (Ph.D.\ thesis, Department of Operations Research, Stanford University, 1981) proved that every matrix in $U \cap Q_0$ is a $P_0$-matrix and conjectured that the same conclusion holds for the larger class $E_0^f \cap Q_0$ of fully semimonotone $Q_0$-matrices. Murthy and Parthasarathy [SIAM J.\ Matrix Anal.\ Appl....
Sajal Ghosh· 0 citations
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