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Mathematics by Experiment: Plausible Reasoning in the 21st Century, extended second edition, A K
 2008. EXPERIMENTATION AND COMPUTATION 19
, 2008
"... If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elab ..."
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If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elaborated in references [1–7]. 1 I shall generally explore experimental and heuristic mathematics and give (mostly) accessible, primarily visual and symbolic, examples. The emergence of powerful mathematical computing environments like Maple and Matlab, the growing
Overview of some general results in combinatorial enumeration
, 2008
"... This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part ..."
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This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of contextfree languages, 3. holonomicity (i.e., Precursiveness) of numbers of labeled regular graphs, 4. frequent occurrence of the asymptotics cn −3/2 r n and 5. ultimate modular periodicity of numbers of MSOLdefinable structures. 1
On some subgroups of the multiplicative group of finite rings
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX
, 2004
"... Let S be a subset of Fq, the field of q elements and h ∈ Fq[x] a polynomial of degree d> 1 with no roots in S. Consider the group generated by the image of {x − s  s ∈ S} in the group of units of the ring Fq[x]/(h). In this paper we present a number of lower bounds for the size of this group. O ..."
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Let S be a subset of Fq, the field of q elements and h ∈ Fq[x] a polynomial of degree d> 1 with no roots in S. Consider the group generated by the image of {x − s  s ∈ S} in the group of units of the ring Fq[x]/(h). In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective line over finite fields.
Applications Of The Classical Umbral Calculus
 Algebra Universalis
"... We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences. 1. ..."
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We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences. 1.
THE PERIOD OF THE BELL NUMBERS MODULO A PRIME
, 2010
"... Abstract. We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np =(p p − 1)/(p − 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investiga ..."
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Abstract. We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np =(p p − 1)/(p − 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investigation leads to interesting new results about the possible prime factors of Np. For example, we show that if p is an odd positive integer and m is a positive integer and q =4m 2 p + 1 is prime, then q divides p m2 p − 1. Then we explain how this theorem influences the probability that q divides Np. 1.
On psiumbral extension of Stirling numbers and Dobinskilike formulas
 Advan. Stud. Contemp. Math
"... ψumbral extensions of the Stirling numbers of the second kind are considered and the resulting new type of Dobinskilike formulas are discovered. These extensions naturally encompass the two well known qextensions.The further consecutive ψumbral extensions of CarlitzGould qStirling numbers are ..."
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ψumbral extensions of the Stirling numbers of the second kind are considered and the resulting new type of Dobinskilike formulas are discovered. These extensions naturally encompass the two well known qextensions.The further consecutive ψumbral extensions of CarlitzGould qStirling numbers are therefore realized here in a twofold way. The fact that the umbral qextended Dobinski formula may also be interpreted as the average of powers of random variable Xq with the qPoisson distribution singles out the qextensions which appear to be a kind of ”bifurcation point ” in the domain of ψumbral extensions as expressed by Observations 2.1 and 2.2 as tackled with the paper closing down question.
ON A CONJECTURE OF WILF
"... Abstract. Let n and k be natural numbers and let S(n, k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum nX (−1) j S(n, j) j=0 is nonzero for all n> 2. We prove this conjecture for all n � ≡ 2 and � ≡ 2944838 mod 3145728 and discuss applications ..."
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Abstract. Let n and k be natural numbers and let S(n, k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum nX (−1) j S(n, j) j=0 is nonzero for all n> 2. We prove this conjecture for all n � ≡ 2 and � ≡ 2944838 mod 3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of padic series. 1.
Approximate constructions in finite fields
 Proc. 3rd Conf. on Finite Fields and Appl
, 1995
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NonVanishing of UppuluriCarpenter Numbers
, 2006
"... The Bell numbers count the number of set partitions of {1,..., n}. They are the integer coefficients Bn in � ∞ tn ..."
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The Bell numbers count the number of set partitions of {1,..., n}. They are the integer coefficients Bn in � ∞ tn
CLOSED EXPRESSIONS FOR AVERAGES OF SET PARTITION STATISTICS
"... Abstract. In studying the enumerative theory of super characters of the group of upper triangular matrices over a finite field we found that the moments (mean, variance and higher moments) of novel statistics on set partitions of [n] = {1, 2, · · · , n} have simple closed expressions as linear c ..."
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Abstract. In studying the enumerative theory of super characters of the group of upper triangular matrices over a finite field we found that the moments (mean, variance and higher moments) of novel statistics on set partitions of [n] = {1, 2, · · · , n} have simple closed expressions as linear combinations of shifted bell numbers. It is shown here that families of other statistics have similar moments. The coefficients in the linear combinations are polynomials in n. This allows exact enumeration of the moments for small n to determine exact formulae for all n. 1.