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This question is motivated by the comments and the answer to this MO-question. First let us recall some definitions. A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\...

I'm looking for combinatorial proofs of the convolutional identity COP below and its specializations I) and II). (Edit 6/2/2023: A combinatorial proof is sketched in a blog post by Mike Spivey of a ...

The presentation given in Wikipedia for the infinite dihedral group is $$\langle r,s\mid s^2 =1, srs = r^{-1}\rangle.$$ Let $[R]$ denote the infinite set of reciprocal partition polynomials $R_n(u_1,...

Let's consider the sequence of coefficients of $\prod_{i}\frac {1-x^{d_i}} {1-x}$, where $d_i$ is a monotonically increasing nonnegative integer sequence. How to prove that the coefficients form an ...