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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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