Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture

Jean-Marie de Koninck; Nicolas Doyon; William Verreault

Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture

Number Theory 2021-07-21 v1

Authors: Jean-Marie de Koninck , Nicolas Doyon , William Verreault

Abstract

Given two integers $k\geq 2$ and $a>1$ , let $N_k(a)$ stand for the number of multinomial coefficients, with $k$ terms, equal to $a$ . We study the behavior of $N_k(a)$ and show that its average and normal orders are equal to $k(k-1)$ . We also prove that $N_k(a)=O\left((\log a/\log\log a)^{k-1}\right)$ and make several propositions about extreme results regarding large values of $N_k(a)$ .

Keywords

multinomial coefficients singmaster's conjecture average order normal order extreme values number theory

Cite

@article{arxiv.2107.09107,
  title  = {Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture},
  author = {Jean-Marie de Koninck and Nicolas Doyon and William Verreault},
  journal= {arXiv preprint arXiv:2107.09107},
  year   = {2021}
}

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11 pages

Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture

Abstract

Keywords

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