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ℓ-ADIC 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 ℓ-adically to 0. Although these congruences have inspired research in many directions, little is known about the ℓ-adic 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 ℓ-adically to 0. Although these congruences have inspired research in many directions, little is known about the ℓ-adic 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 ℓ-adically converge to linear combinations of at most ⌊ ℓ−1 12 ⌋− ⌊ ℓ2−1 24ℓ ⌋ many special q-series. 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 p-adic 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 half-integral 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 half-integral 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 Delta-function.
Non-Vanishing of Uppuluri-Carpenter 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.”
