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On the number of triangulation simplexes. (English. Russian original) Zbl 0837.05007
Russ. Acad. Sci., Izv., Math. 44, No. 1, 1-20 (1995); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 58, No. 1, 3-21 (1994).
This paper is devoted to an interesting generalization of the multiplicativity of the Euler characteristic, so preparing some possible further developments. Consider the set of ‘$$p$$-simplicial polynomials’ $${x \choose p} = {1 \over p!} (x)_p$$, $$p = 0,1,2, \dots$$ as a basis of the linear $$\mathbb{Q}$$-space $$\mathbb{Q} [x]$$ of all polynomials with rational coefficients. The starting point here is the following Theorem 1: Given any simplexes $$\sigma_{m_1}, \sigma_{m_2}, \dots, \sigma_{m_s}$$, the binomial generating function $$F_0 (x) = \sum_{p \geq 0} \alpha_p {x \choose p}$$ for the numbers $$\alpha_p$$ of internal $$p$$-simplexes of a standardly triangulated product $$\sigma_{m_1} \times \sigma_{m_2} \times \cdots \times \sigma_{m_s}$$ is given by the rule $$F_0 (x) = {x \choose m_1} {x \choose m_2} \cdot \dots \cdot {x \choose m_s}$$. Firstly, this rule is generalized by the author to give the binomial generating function of the numbers $$\alpha_p$$ for standardly triangulated products of triangulated convex polyhedra $$\Pi$$ and $$\Sigma$$ (or, even, for the case, where $$\Pi$$ and $$\Sigma$$ are pseudovarieties in the sense of the book [P. S. M. Aleksandrov, Introduction to homological dimension of general combinatorial topology, Moscow (1975; Zbl 0441.55002)]. Namely, it is shown (Theorems 1A (and 1B)) that it holds $$F_0 (\Pi \times \Sigma) = F_0 (\Pi) F_0 (\Sigma)$$; here $$F_0 (\Pi)$$ is defined as the series $$\sum_{p \geq 0} \alpha_p (\Pi) {x \choose p}$$, with $$\alpha_p (\Pi)$$ being the number of $$p$$-simplexes having nonempty intersection with the interior of the underlying space $$|\Pi |$$ of triangulation of $$\Pi$$. Secondly, this multiplicativity rule is generalized to give $$F(\Pi \times \Sigma) = F(\Pi) \cdot F (\Sigma)$$ for the series $$F(\Pi) = F(\Pi; x,t) = \sum_{p,q \geq 0} \alpha^q_p (\Pi) {x \choose p} t^q$$, where $$\alpha^q_p (\Pi)$$ denotes the number of those $$p$$-simplexes for the triangulation of the convex polyhedron $$\Pi$$ that are internal for a face having codimension $$q$$. Note that $$\alpha^0_p (\Pi) = \alpha_p (\Pi)$$ and $$F(\Pi; x,0) = F_0 (\Pi)$$. Exponential generating functions of the type $v(\Pi, \Sigma; x,t) = \sum_{p,q} \alpha_p (\Pi^q \times \Sigma) x^p t^q/p!q! = \sum_q E(\Pi^q \times \Sigma ;x) t^q/q!$ with $$E(\Pi; x)$$ defined as the series $$\sum_p \alpha_p (\Pi) {x^p \over p!}$$, are also considered. It is proved (Theorem 3) that this series $$v$$ satisfies the evolutionary differential equation $$v_t' = \widetilde E (\Pi; x({\partial \over \partial x} + 1))v$$; here, for a polynomial $$A = \sum_k a_k x^k$$, the operator $$\sum_k a_k x^k({\partial \over \partial x} + 1)^k$$ is denoted by $$\widetilde A (x ({\partial \over \partial x} + 1))$$. The most intriguing part of this useful paper deals with congruences involving the numbers $$\alpha_p (\sigma_{m_1} \times \sigma_{m_2} \times \cdots \times \sigma_{m_s})$$, generalizing the congruences in the von Staudt theorem, concerning Bernoulli numbers. The point of view of the calculus of finite differences is used, and the $$q$$-th Bernoulli number $$B_q$$ is represented as the sum $$\sum^{q + 1}_{p = 1} (-1)^{p-1} \alpha_{p - 1} (\sigma^q_1)/p$$ as presented in the book [A. O. Gel’fond, Calculus of finite differences (authorized English translation of the 3rd Russian edition, Moscow, 1967) Delhi (1971; Zbl 0264.39001)].
##### MSC:
 05A10 Factorials, binomial coefficients, combinatorial functions 05A15 Exact enumeration problems, generating functions 54E99 Topological spaces with richer structures
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