An identity relating $n$-nacci numbers, partitions, and products of binomial coefficients
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909999906684928 |
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| author | Dragutinović, Dušan |
| author_facet | Dragutinović, Dušan |
| contents | We study the combinatorial properties of final types, which are certain non-decreasing sequences of integers, together with the partitions naturally associated with them. As a consequence, we obtain an identity expressing the $n$-nacci numbers as sums of products of binomial coefficients over these partitions, generalizing the classical identity for $n = 2$ that expresses Fibonacci numbers in this way. We also examine how the partial order on the set of all partitions of a fixed integer induced by the ordering of final types compares with two natural partial orders on the same set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17401 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An identity relating $n$-nacci numbers, partitions, and products of binomial coefficients Dragutinović, Dušan Combinatorics Number Theory We study the combinatorial properties of final types, which are certain non-decreasing sequences of integers, together with the partitions naturally associated with them. As a consequence, we obtain an identity expressing the $n$-nacci numbers as sums of products of binomial coefficients over these partitions, generalizing the classical identity for $n = 2$ that expresses Fibonacci numbers in this way. We also examine how the partial order on the set of all partitions of a fixed integer induced by the ordering of final types compares with two natural partial orders on the same set. |
| title | An identity relating $n$-nacci numbers, partitions, and products of binomial coefficients |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2601.17401 |