Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem

Ball, Tyler; Edgar, Tom; Juda, Daniel

doi:10.4169/math.mag.87.2.135

Cited by 12 publications

(17 citation statements)

References 4 publications

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“…. , n k ) m , the product k i=0 (n i + 1) counts the number of integers dominated by n (see [4]). We will use the interpretation of the product in terms of the m-dominance order in what follows.…”

Section: Preliminaries and Statement Of The Main Resultsmentioning

confidence: 99%

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The distribution of the number of parts of $m$-ary partitions modulo $m$

Edgar¹

2017

Rocky Mountain J. Math.

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We investigate the number of parts modulo m of m-ary partitions of a positive integer n. We prove that the number of parts is equidistributed modulo m on a special subset of m-ary partitions. As consequences, we explain when the number of parts is equidistributed modulo m on the entire set of partitions, and we provide an alternate proof of a recent result of Andrews, Fraenkel, and Sellers about the number of m-ary partitions modulo m.

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Section: Preliminaries and Statement Of The Main Resultsmentioning

confidence: 99%

“…As previously mentioned, there are k i=1 (n i + 1) integers equivalent to n mod m that are m-dominated by n (see [4] and use the fact that b is equivalent n mod m if and only if b 0 = n 0 ). Thus, we see that…”

Section: Proof Of Theorem 12 and Consequencesmentioning

confidence: 99%

The distribution of the number of parts of $m$-ary partitions modulo $m$

Edgar¹

2017

Rocky Mountain J. Math.

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“…Combining Corollary 9 with Proposition 1 shows that in fact L B (q) (multiplied by B − 1) represents the generating function for the correction term c B (n − 1, 1). 6 Corollary 10. We have for B ≥ 2 and |q| < 1 that…”

Section: Corollarymentioning

confidence: 99%

Digit sums and generating functions

Schneider

2020

Ramanujan J

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We connect a primitive operation from arithmetic -summing the digits of a base-B integer -to q-series generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined with distinctive classes of "Bary" Lambert series, which themselves enjoy nice transformations. We also consider digit sum Dirichlet series.1 Proposition 1 is equivalent to Theorem 2 in [6]; see also [13].

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“…Throughout this paper we assume that b is an integer greater than 1. We begin by introducing the notion of digital dominance as defined in [1] (see also [7]).…”

Section: Sierpinski Matrices Of Sheffer Typementioning

confidence: 99%

“…One generalization of the binomial theorem, due to Callan [2] (see also [6]), expresses the exponents appearing in (1) in terms of the binary sum-of-digits function s(m):…”

Section: Introductionmentioning

confidence: 99%