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Fix a column vector $\pmb{v}$ and consider the cross product $\pmb{v}^T\times\pmb{x}^T$ for any column vector $\pmb{x}\in\mathbb{R}^3$. One can write $$\pmb{v}^T\times\pmb{x}^T=A(\pmb{v})\pmb{x}$$ for …

One of the delights in mathematical research is that some (mostly deep) results in one area remain unknown to mathematicians in other areas, but later, these discoveries turn out to be equivalent! Th …

This question is motivated by Richard Stanley's answer to this MO question. Let $g(n)$ be the number of distinct monomials in the expansion of the determinant of an $n\times n$ generic "skew-symmetri …

For a fixed integer $N\in\mathbb{N}$ consider the multi-set $A_2(N)$ of decimal digits of $2^n$, for $n=1,2,\dots,N$. For example, $$A_2(8)=\{2,4,8,1,6,3,2,6,4,1,2,8,2,5,6\}.$$ Similarly, define the m …

I have been pondering about certain conjectures and theorems viewed as either low vs high dimensions, or smaller vs larger primes, or anything of the sort "low vs high order". Let me mention a couple …

Let $\mathcal{A}_n$ be the set of all Alternating Sign Matrices (ASM) of size $n\times n$. The cardinality $\#\mathcal{A}_n$ is well-known $$\#\mathcal{A}_n=\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!}.$$ …

Consider the finite sums $$F_n(q)=\sum_{k=1}^nq^{\binom{k}2}$$ with exponents the triangular numbers $\binom{k}2$. When $n$ is odd, it appears that $F_n(q)$ does not factorize over $\mathbb{Z}[q]$. On …

The Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$ have the generating function $$c(x)=\frac{1-\sqrt{1-4x}}{2x}.$$ Let $a\in\mathbb{R}^+$. It seems that the following holds true $$\frac{c(x)^a}{\sqrt{1- …

It's known that Mark E. Walker proved the "weaker" version of Buchsbaum-Eisenbud-Horrocks' Conjecture (BEH). Although the claim was stated to hold in arbitrary field $k$, Walker's proof does not seem …

Fix an integer $t\geq0$ and consider the sequence $T_{0,t}=1$ and for $n\geq1$, with $k*=n-1-k+t\mod n$, $$T_{n,t}=\sum_{k=0}^{n-1}T_{k,t}T_{k*,t}.$$ EXAMPLES. Some initial terms: (a) the case $t=0$: …

Let $t\in\Bbb{N}$ and consider the sequences $p_t(n)$ defined by $$\sum_{n\geq0}p_t(n)x^n=\prod_{i\geq1}\frac1{(1-x^i)^t}=(x;x)_{\infty}^{-t}.$$ The numbers $p_t(n)$ can be regarded as enumerating par …

Among the families of sequences studied by Nicolaas de Bruijn (Asymptotic Methods in Analysis, 1958), let's focus on the (modified) $$\hat{S}(4,n)=\frac1{n+1}\sum_{k=0}^{2n}(-1)^{n+k}\binom{2n}k^4.$$ …

Assume $a\in\mathbb{N}$. Among the families of sequences studied by Nicolaas de Bruijn (Asymptotic Methods in Analysis, 1958), let's focus on the (modified) $$\hat{S}(2a,n)=\frac1{n+1}\sum_{k=0}^{2n}( …

Let $\lambda\vdash n$ denote a partition $\lambda$ of $n$ and let $\square\in\lambda$ denote a box $\square$ in the Young diagram of $\lambda$. QUESTION. Can you list a pair of (distinct) statistics …

This is a follow up on my earlier MO post. Given $\lambda$ an integer partition of $n$, let $h_{ij}(\lambda)$ denote the hook length of cell $(i,j)$ in the Young diagram of $\lambda$. Let $$f_n=\sum_{ …