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ℓADIC PROPERTIES OF THE PARTITION FUNCTION
"... Abstract. Ramanujan’s famous partition congruences modulo powers of 5, 7, and 11 imply that certain sequences of partition generating functions tend ℓadically to 0. Although these congruences have inspired research in many directions, little is known about the ℓadic behavior of these sequences for ..."
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Abstract. Ramanujan’s famous partition congruences modulo powers of 5, 7, and 11 imply that certain sequences of partition generating functions tend ℓadically to 0. Although these congruences have inspired research in many directions, little is known about the ℓadic behavior of these sequences for primes ℓ ≥ 13. We show that these sequences are governed by “fractal” behavior. Modulo any power of a prime ℓ ≥ 5, these sequences of generating functions ℓadically converge to linear combinations of at most ⌊ ℓ−1 12 ⌋− ⌊ ℓ2−1 24ℓ ⌋ many special qseries. For ℓ ∈ {5, 7, 11} we have ⌊ ℓ−1 12 ⌋− ⌊ ℓ2−1 24ℓ ⌋ = 0, thereby giving a conceptual explanation of Ramanujan’s congruences. We use the general result to reveal the theory of “multiplicative partition congruences ” that Atkin anticipated in the 1960s. His results and observations are examples of systematic infinite families of congruences which exist for all powers of primes 13 ≤ ℓ ≤ 31 since ⌊ ℓ−1
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.
THE PARTITION FUNCTION AND HECKE OPERATORS
"... Abstract. The theory of congruences for the partition function p(n) depends heavily on the properties of halfintegral weight Hecke operators. The subject has been complicated by the absence of closed formulas for the Hecke images P (z)  T (ℓ2), where P (z) is the relevant modular generating funct ..."
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Abstract. The theory of congruences for the partition function p(n) depends heavily on the properties of halfintegral weight Hecke operators. The subject has been complicated by the absence of closed formulas for the Hecke images P (z)  T (ℓ2), where P (z) is the relevant modular generating function. We obtain such formulas using Euler’s Pentagonal Number Theorem and the denominator formula for the Monster Lie algebra. As a corollary, we obtain congruences for certain powers of Ramanujan’s Deltafunction.
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
© Hindawi Publishing Corp. PRODUCT PARTITIONS AND RECURSION FORMULAE
, 2003
"... Utilizing a method briefly hinted in the author’s paper written in 1991 jointly with V. C. Harris, we derive here a number of unpublished recursion formulae for a variety of product partition functions which we believe have not been considered before in the literature. These include the functionsp∗( ..."
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Utilizing a method briefly hinted in the author’s paper written in 1991 jointly with V. C. Harris, we derive here a number of unpublished recursion formulae for a variety of product partition functions which we believe have not been considered before in the literature. These include the functionsp∗(n;k,h) (which stands for the number of product partitions ofn> 1 into k parts of which h are distinct), and p∗(d)(n;m) (which stands for the number of product partitions of n into exactly m parts with at most d repetitions of any part). We also derive recursion formulae for certain product partition functions without the use of generating functions. 2000 Mathematics Subject Classification: 11P81, 11P82. 1. Introduction. This paper is prompted (and provoked) by a remark made by Kim and Hahn in the introduction of their paper [11] that appeared in this journal a few years back. They said: “we find recursive formulae for the multipartite function p(n1,...,nj). The most useful formula known to this day for actual evaluation of the multipartite partition function is presented in Theorem 4.”