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$P_n$ is given as $$P_n(f) = \sum_{\lambda \,\vdash\, n} (-1)^{\lambda_1} \prod \binom{\lambda_j}{\lambda_{j+1}} f_{j}^{\lambda_j}$$ where the sum is over partitions $\lambda = \lambda_1 \ge \lambda_ …

If we take a general sequence of (unsigned) Comtet numbers of the first kind [1, 2] $$c(n, k) = e_{n-k}(\xi_1, \ldots, \xi_n)$$ then the $n \times n$ submatrix with a row offset of $k$ has determinant …

The Stirling numbers of the first kind satisfy $x^{\underline{n}} = \sum_{k=0}^n s_1(n,k)x^k$. For $n > 0$ we have $s_1(n, 0) = 0$, $s_1(n, 1) = (-1)^{n-1}(n-1)!$, $s_1(n, n) = 1$. If $n > 1$ then $1^ …

$R(n, k, -\tfrac k2)$ is just the central factorial number $T(n, k)$. (Given the definition of the central factorial numbers, it may be more natural to use them in your context than $R$). Consider A13 …

There is some $N(k, p)$ such that $n \ge N(k, p) \implies s(n, k) \equiv 0 \pmod p$. Proof is straightforward by fixing $p$ and using induction on $k$ via the recurrence $$s(n, k) = s(n-1, k-1) - (n-1 …