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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.12328 |
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| _version_ | 1866917209268289536 |
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| author | Chen, Yanru Fu, Houshan Liang, Weikang Wang, Suijie |
| author_facet | Chen, Yanru Fu, Houshan Liang, Weikang Wang, Suijie |
| contents | In this paper, we introduce the bivariate exponential generating function $F_l(x,y)$ for the number of level-$l$ faces of an exponential sequence of arrangements (ESA), and establish the formula $F_l(x,y)=\big(F_1(x,y)\big)^l$ with a combinatorial interpretation. Its specialization at $x=0$ recovers a result first obtained by Chen et al. [3,4] for certain classic ESAs and later generalized to all ESAs by Southerland et al. [8]. As a byproduct, we obtain that an alternating sum of the number of level-$l$ faces is invariant with respect to the choice of ESA, and is exactly the Stirling number of the second kind. We also extend the binomial-basis expansion theorem [3,4,14] and Stanley's formula on ESAs [9] from characteristic polynomials to Whitney polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12328 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Level of Faces for Exponential Sequence of Arrangements Chen, Yanru Fu, Houshan Liang, Weikang Wang, Suijie Combinatorics In this paper, we introduce the bivariate exponential generating function $F_l(x,y)$ for the number of level-$l$ faces of an exponential sequence of arrangements (ESA), and establish the formula $F_l(x,y)=\big(F_1(x,y)\big)^l$ with a combinatorial interpretation. Its specialization at $x=0$ recovers a result first obtained by Chen et al. [3,4] for certain classic ESAs and later generalized to all ESAs by Southerland et al. [8]. As a byproduct, we obtain that an alternating sum of the number of level-$l$ faces is invariant with respect to the choice of ESA, and is exactly the Stirling number of the second kind. We also extend the binomial-basis expansion theorem [3,4,14] and Stanley's formula on ESAs [9] from characteristic polynomials to Whitney polynomials. |
| title | Level of Faces for Exponential Sequence of Arrangements |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.12328 |