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| Formato: | Preprint |
| Publicado: |
2021
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| Acceso en línea: | https://arxiv.org/abs/2104.12445 |
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| _version_ | 1866909103519956992 |
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| author | Santocanale, Luigi |
| author_facet | Santocanale, Luigi |
| contents | Let $\Bigl\langle\matrix{n\cr k}\Bigr\rangle$, $\Bigl\langle\matrix{B_n\cr k}\Bigr\rangle$, and $\Bigl\langle\matrix{D_n\cr k}\Bigr\rangle$ be the Eulerian numbers in the types A, B, and D, respectively -- that is, the number of permutations of n elements with $k$ descents, the number of signed permutations (of $n$ elements) with $k$ type B descents, the number of even signed permutations (of $n$ elements) with $k$ type D descents. Let $S_n(t) = \sum_{k = 0}^{n-1} \Bigl\langle\matrix{n\cr k}\Bigr\rangle t^k$, $B_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{B_n\cr k}\Bigr\rangle t^k$, and $D_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{D_n\cr k}\Bigr\rangle t^k$. We give bijective proofs of the identity $$B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2^n tS_n(t^2)$$ and of Stembridge's identity $$D_n(t) = B_n(t) - n2^{n-1}tS_{n-1}(t).$$ These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs $(w, E)$ with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$, which we use to obtain bijective proofs of enumerative results for threshold graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_12445 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Bijective proofs for Eulerian numbers of types B and D Santocanale, Luigi Logic in Computer Science Logic Let $\Bigl\langle\matrix{n\cr k}\Bigr\rangle$, $\Bigl\langle\matrix{B_n\cr k}\Bigr\rangle$, and $\Bigl\langle\matrix{D_n\cr k}\Bigr\rangle$ be the Eulerian numbers in the types A, B, and D, respectively -- that is, the number of permutations of n elements with $k$ descents, the number of signed permutations (of $n$ elements) with $k$ type B descents, the number of even signed permutations (of $n$ elements) with $k$ type D descents. Let $S_n(t) = \sum_{k = 0}^{n-1} \Bigl\langle\matrix{n\cr k}\Bigr\rangle t^k$, $B_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{B_n\cr k}\Bigr\rangle t^k$, and $D_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{D_n\cr k}\Bigr\rangle t^k$. We give bijective proofs of the identity $$B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2^n tS_n(t^2)$$ and of Stembridge's identity $$D_n(t) = B_n(t) - n2^{n-1}tS_{n-1}(t).$$ These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs $(w, E)$ with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$, which we use to obtain bijective proofs of enumerative results for threshold graphs. |
| title | Bijective proofs for Eulerian numbers of types B and D |
| topic | Logic in Computer Science Logic |
| url | https://arxiv.org/abs/2104.12445 |