Preliminary Sermon: Humans will be Humans; The Medium is the Message The famous Catalan numbers (see [Sl1]), count zillions of combinatorial families (see [St]) and many humans have fun trying to find ‘nice’ bijections between family A and family B. While this may be fun for a while, sooner or later this game gets old, especially since the real reason Catalan numbers are so ubiquitous is their ...

Let Ng(f ) denote the number of rooted maps of genus g having f edges. An exact formula for Ng(f ) is known for g = 0 (Tutte, 1963), g = 1 (Arques, 1987), g = 2,3 (Bender and Canfield, 1991). In the present paper we derive an enumeration formula for the number Θγ (e) of unrooted maps on an orientable surface Sγ of a given genus γ and with a given number of edges e. It has a form of a linear com...

Let Ng(f) denote the number of rooted maps of genus g having f edges. Exact formula for Ng(f) is known for g = 0 (Tutte 1963), g = 1 (Arques 1987), g = 2, 3 (Bender and Canfield 1991). In the present paper we derive an enumeration formula for the number Θγ(e) of unrooted maps on an orientable surface Sγ of given genus γ and given number of edges e. It has a form of a linear combination ∑ i,j ci...

An investigation is made of the polynomials fk(n) = ,S(n + I?, n) and g&z) == (l)k s(n, n k), where s and s denote the Stirling numbers of the second and first kind, respectively. The main result gives a combinatorial interpretation of the coefficients of the polynomial (1 x)~~+* .Z"t&(n)xn analogous to the well-known combinatorial interpretation of the Eulerian numbers in terms of descents of ...

In this article, we formalize in Mizar [7] the definition of “torsion part” of Z-module and its properties. We show Z-module generated by the field of rational numbers as an example of torsion-free non free Z-modules. We also formalize the rank-nullity theorem over finite-rank free Z-modules (previously formalized in [1]). Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lo...

and Applied Analysis 3 The purpose of this paper is to derive a new concept of higher-order q-Bernoulli numbers and polynomials with weight α from the fermionic p-adic q-integral on Zp. Finally, we present a systemic study of some families of higher-order q-Bernoulli numbers and polynomials with weight α. 2. Higher Order q-Bernoulli Numbers with Weight α Let β ∈ Z and α ∈ N in this paper. For k...

Throughout this paper, we always make use of the following notation: N = {1, 2, 3, . . .} denotes the set of natural numbers, N0 = {0, 1, 2, 3, . . .} denotes the set of nonnegative integers, Z−0 = {0,−1,−2,−3, . . .} denotes the set of nonpositive integers, Z denotes the set of integers, R denotes the set of real numbers, C denotes the set of complex numbers. The generalized Bernoulli polynomi...

Let p be a fixed odd prime number. Throughout this paper, we always make use of the following notations: Z denotes the ring of rational integers, Zp denotes the ring of padic rational integer, Qp denotes the ring of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N {0}. Let Cpn {ζ | ζpn 1} be the cyclic g...

1. BALANCING NUMBERS We call an Integer n e Z a balancing number if 1+ 2+ --+ (»l ) = (w + l) + (w + 2) +••• + (» + >•) (1) for some r e Z. Here r is called the balancer corresponding to the balancing number n. For example, 6, 35, and 204 are balancing numbers with balancers 2, 14, and 84, respectively. It follows from (1) that, if n is a balancing number with balancer r, then n2^(n + r)(n + r ...