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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2501.07692 |
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| _version_ | 1866909455544745984 |
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| author | Sagan, Bruce E. |
| author_facet | Sagan, Bruce E. |
| contents | The Euler numbers have been widely studied. A signed version of the Euler numbers of even subscript are given by the coefficients of the exponential generating function 1/(1+x^2/2!+x^4/4!+...). Leeming and MacLeod introduced a generalization of the Euler numbers depending on an integer parameter d where one takes the coefficients of the expansion of 1/(1+x^d/d!+x^{2d}/(2d)!+...). These numbers have been shown to have many interesting properties despite being much less studied. And the techniques used have been mainly algebraic. We propose a combinatorial model for them as signed sums over ordered partitions. We show that this approach can be used to prove a number of old and new results including a recursion, integrality, and various congruences. Our methods include sign-reversing involutions and Möbius inversion over partially ordered sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07692 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Euler numbers and ordered set partitions Sagan, Bruce E. Number Theory Combinatorics 11B68 (Primary) 05A18, 11P83 (Secondary) The Euler numbers have been widely studied. A signed version of the Euler numbers of even subscript are given by the coefficients of the exponential generating function 1/(1+x^2/2!+x^4/4!+...). Leeming and MacLeod introduced a generalization of the Euler numbers depending on an integer parameter d where one takes the coefficients of the expansion of 1/(1+x^d/d!+x^{2d}/(2d)!+...). These numbers have been shown to have many interesting properties despite being much less studied. And the techniques used have been mainly algebraic. We propose a combinatorial model for them as signed sums over ordered partitions. We show that this approach can be used to prove a number of old and new results including a recursion, integrality, and various congruences. Our methods include sign-reversing involutions and Möbius inversion over partially ordered sets. |
| title | Generalized Euler numbers and ordered set partitions |
| topic | Number Theory Combinatorics 11B68 (Primary) 05A18, 11P83 (Secondary) |
| url | https://arxiv.org/abs/2501.07692 |