Timeline for (Elementary?) combinatorial identity expressing binomial coefficients as an alternating sum over permutations.
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2020 at 16:16 | comment | added | Tom Copeland | Additional info: Rota and others have discussed the combinatorics of $$n! \; \binom{x}{n}=ST1_n(x)=\sum_{k=0}^n ST1_{n,k} \; x^n= (-1)^n! \binom{-x-1+n}{n} $$, the classic Stirling polynomials of the first kind, a.k.a. the falling factorials, which easily generalize to the cycle index partition polynomials of the symmetric groups. The "inverse/reciprocal" polynomials are the Stirling polynomials of the second kind, which generalize to the Bell/Faa di Bruno partition polynomials of the much touted Faa di Bruno Hopf algebra--all central characters in combinatorics, algebra, and analysis. | |
| Jun 30, 2010 at 22:59 | vote | accept | MTS | ||
| Jun 30, 2010 at 22:49 | answer | added | Qiaochu Yuan | timeline score: 16 | |
| Jun 30, 2010 at 22:25 | history | asked | MTS | CC BY-SA 2.5 |