Stirling operators in spatial combinatorics

We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol $(m)_n$ can be extended from natural number $m\in\mathbb N$ to falling factorials $(z)_n=z(z-1)\dotsm (z-n+1)$ an argument $z$ $\mathbb F=\mathbb R\text{ or }\mathbb C$, numbers first second kinds are coefficients expansions $(z)_n$ through $z^k$, $k\leq n$ vice versa. When taking into account positions elements in locally compact Polish space $X$, we replace by configurations -- discrete Radon measures $\gamma=\sum_i\delta_{x_i}$ on where $\delta_{x_i}$ is Dirac measure with mass at $x_i$.The $(\gamma)_n:=\sum_{i_1}\sum_{i_2\ne i_1}\dotsm\sum_{i_n\ne i_1,\dots, i_n\ne i_{n-1}}\delta_{(x_{i_1},x_{i_2},\dots,x_{i_n})}$ naturally mappings $M^{(1)}(X)\ni\omega\mapsto (\omega)_n\in M^{(n)}(X)$, $M^{(n)}(X)$ denotes F$-valued, symmetric (for $n\ge2$) $X^n$. There duality between $\mathcal {CF}^{(n)}(X)$ continuous functions $X^n$ support. The operators kind, $\mathbf{s}(n,k)$ $\mathbf{S}(n,k)$, linear operators, acting spaces {CF}^{(k)}(X)$ such that their dual $M^{(k)}(X)$ $M^{(n)}(X)$, satisfy $(\omega)_n=\sum_{k=1}^n\mathbf{s}(n,k)^*\omega^{\otimes k}$ $\omega^{\otimes n}=\sum_{k=1}^n\mathbf{S}(n,k)^*(\omega)_k$, respectively. derive combinatorial properties present connections generalization Poisson point process Wick ordering under canonical commutation relations.