69 problems

Let $ped(n)$ denote the number of partitions of $n$ in which even parts are distinct and odd parts are unrestricted. Let $p\geq5$ be a prime satisfying … and let $t$ be a positive…

Let $p_r(n)$ denote the coefficients in the partition-generating series used in the paper. For a positive integer $\lambda$, let $\mu_{\lambda}$ and $\delta_{\lambda}$ satisfy … Hi…

Density conjecture. The natural density of indices $m$ for which $p_0(m)$ is even exists and equals

Mod 4 congruence conjecture. For every $n\geq0$ and every integer $j$ with $0\leq j\leq\ell-2$,

Further generalized cubic partition congruence conjecture.

Let $n\geq 0$, $i\geq 3$, and $k\geq 0$. The overcubic partition tuple function $\overline{b}_m(r)$ is considered at the indicated arithmetic progressions. Congruence conjecture. F…

Let $m$ be a positive integer with $m\equiv 1\pmod{24}$. Suppose that all prime divisors of $m$ are congruent to $1$, $5$, $7$, or $11$ modulo $24$, and write … where…

Let an overcolored partition of $n$ be a partition in which even parts may appear in one of $r$ colors and odd parts may appear in one of $s$ colors, with the first occurrence of e…

Let $d_k(n)$ denote the $k$-elongated plane partition function, and let $c,n\geq 0$. For a list of integers in an argument, interpret the corresponding congruence as holding for ea…

Let $b_m(n)$ be the coefficients of the generating function for $m$-regular partitions. For even $m$, write … where $m_0$ is odd. The odd density of a sequence is the limiting prop…

Let $m_0$ be the odd part of an even integer $m$, and let $b_m(n)$ denote the coefficients of the corresponding $m$-regular partition generating function. Theorem gives congruences…

Let $b_{3,e}(n)$ and $b_{3,o}(n)$ denote the numbers of $3$-regular partitions of $n$ having, respectively, an even and an odd number of parts. Let $n\geqslant 0$, let…

Noncongruence and odd-density conjecture. The following assertions hold:

Banerjee–Bringmann–Bachraoui's conjecture. For all integers $n$ and $k$,

Dyadic congruence conjecture. For every integer $n\geq 0$, the congruences

Let $DSOME(n)$ denote the sum of all odd parts in the partitions of $n$ into distinct parts minus the sum of all even parts. DSOME congruence conjecture. For all integers $n\geq0$,…

Das et al.'s conjecture. For all $n\geq 0$ and $i\geq 1$, these five congruences hold.

Keith's conjecture. For $n\geq 0$,

Let $M_{C1}(m,n)$ and $M_{C5}(m,n)$ be the coefficients defined by the corresponding two-variable generating functions in the source, with $m\in\mathbb{Z}$ and $n\in\mathbb{N}$, wh…

Let $K(n)$ be defined by its generating function … where $f_k=(q^k;q^k)_\infty$ and $(a;q)_\infty=\prod_{i\geq 0}(1-aq^i)$. Dasappa et al.'s conjecture. For all $n\geq 0$ and…

The recurrence congruence conjecture.

Let $\overline{R_{4,9}}(n)$ denote the number of $(4,9)$-regular overpartitions of $n$. For integers $n\geq0$, $\ell\geq2$, and $1\leq k\leq\ell$, Alanazi–Munagi–Saikia's conjectur…

Let $P(N)$ be defined by $P(N)=p(n)$ when $N=24n-1$, and $P(N)=0$ if $N<-1$, if $N\not\equiv -1\pmod{24}$, or if $N$ is nonintegral. Let $\left(\frac{a}{b}\right)$ denote the Jacob…

Saikia and Sarma's conjecture. For $n,\alpha\geq 0$,

Overcubic partition congruence conjecture modulo powers of three.