On the Number of Partitions of an Integer in the $m$-bonacci Base

Marcia Edson[1]; Luca Q. Zamboni[1]

• [1] University of North Texas Department of Mathematics PO Box 311430 Denton, TX 76203-1430 (USA)
• Volume: 56, Issue: 7, page 2271-2283
• ISSN: 0373-0956

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Abstract

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For each $m\ge 2,$ we consider the $m$-bonacci numbers defined by ${F}_{k}={2}^{k}$ for $0\le k\le m-1$ and ${F}_{k}={F}_{k-1}+{F}_{k-2}+\cdots +{F}_{k-m}$ for $k\ge m.$ When $m=2,$ these are the usual Fibonacci numbers. Every positive integer $n$ may be expressed as a sum of distinct $m$-bonacci numbers in one or more different ways. Let ${R}_{m}\left(n\right)$ be the number of partitions of $n$ as a sum of distinct $m$-bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for ${R}_{m}\left(n\right)$ involving sums of binomial coefficients modulo $2.$ In addition we show that this formula may be used to determine the number of partitions of $n$ in more general numeration systems including generalized Ostrowski number systems in connection with Episturmian words.

How to cite

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Edson, Marcia, and Zamboni, Luca Q.. "On the Number of Partitions of an Integer in the $m$-bonacci Base." Annales de l’institut Fourier 56.7 (2006): 2271-2283. <http://eudml.org/doc/10204>.

@article{Edson2006,
abstract = {For each $m\ge 2,$ we consider the $m$-bonacci numbers defined by $F_k=2^k$ for $0\le k\le m-1$ and $F_k=F_\{k-1\} + F_\{k-2\} +\cdots +F_\{k-m\}$ for $k\ge m.$ When $m=2,$ these are the usual Fibonacci numbers. Every positive integer $n$ may be expressed as a sum of distinct $m$-bonacci numbers in one or more different ways. Let $R_m(n)$ be the number of partitions of $n$ as a sum of distinct $m$-bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for $R_m(n)$ involving sums of binomial coefficients modulo $2.$ In addition we show that this formula may be used to determine the number of partitions of $n$ in more general numeration systems including generalized Ostrowski number systems in connection with Episturmian words.},
affiliation = {University of North Texas Department of Mathematics PO Box 311430 Denton, TX 76203-1430 (USA); University of North Texas Department of Mathematics PO Box 311430 Denton, TX 76203-1430 (USA)},
author = {Edson, Marcia, Zamboni, Luca Q.},
journal = {Annales de l’institut Fourier},
keywords = {Numeration systems; Fibonacci numbers; Fine and Wilf theorem; episturmian words; numeration systems; epi-Sturmian words},
language = {eng},
number = {7},
pages = {2271-2283},
publisher = {Association des Annales de l’institut Fourier},
title = {On the Number of Partitions of an Integer in the $m$-bonacci Base},
url = {http://eudml.org/doc/10204},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Edson, Marcia
AU - Zamboni, Luca Q.
TI - On the Number of Partitions of an Integer in the $m$-bonacci Base
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 7
SP - 2271
EP - 2283
AB - For each $m\ge 2,$ we consider the $m$-bonacci numbers defined by $F_k=2^k$ for $0\le k\le m-1$ and $F_k=F_{k-1} + F_{k-2} +\cdots +F_{k-m}$ for $k\ge m.$ When $m=2,$ these are the usual Fibonacci numbers. Every positive integer $n$ may be expressed as a sum of distinct $m$-bonacci numbers in one or more different ways. Let $R_m(n)$ be the number of partitions of $n$ as a sum of distinct $m$-bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for $R_m(n)$ involving sums of binomial coefficients modulo $2.$ In addition we show that this formula may be used to determine the number of partitions of $n$ in more general numeration systems including generalized Ostrowski number systems in connection with Episturmian words.
LA - eng
KW - Numeration systems; Fibonacci numbers; Fine and Wilf theorem; episturmian words; numeration systems; epi-Sturmian words
UR - http://eudml.org/doc/10204
ER -

References

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