Question

Сombinatorial proof for recurrent formula of stirling's numbers

score 82 · Answer 1 · 2020-09-21 12:41:58

82

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Сombinatorial proof for recurrent formula of stirling's numbers

Published on 21 Sep 2020 - 12:41

score 656 · Answer 2 · 2020-09-28 18:21:39

656

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Give combinatorial proof for Stirling number of second kind formula: $S(n,k) = \sum_{i=1}^{n}S(n-i, k-2)(k^{i-1}-(k-1)^{i-1})$

Published on 28 Sep 2020 - 18:21

score 275 · Answer 3 · 2020-10-02 13:33:10

275

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Use combinatorics to prove $L\left(n,k\right)=\sum_{j=0}^{n}{ n\brack j}{j\brace k}$

Published on 02 Oct 2020 - 13:33

score 164 · Answer 4 · 2020-10-04 10:36:58

164

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Using combinatorial argument to prove that ${ n\brack 3}=\frac{1}{2}\left(n-1\right)!\left[\left(H_{n-1}\right)^{2}-H_{n-1}^{\left(2\right)}\right]$

Published on 04 Oct 2020 - 10:36

score 269 · Answer 5 · 2020-10-05 16:24:19

269

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A Stirling number identity

Published on 05 Oct 2020 - 16:24

score 164 · Answer 6 · 2026-03-27 23:14:27

164

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Number of ways to distribute balls in three out of five boxes

Published on 27 Mar 2026 - 23:14

score 669 · Answer 7 · 2020-10-14 16:33:26

669

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Prove that for all positive integers $n$, $\sum_{k=1}^{n}(-1)^kk!S(n,k)=(-1)^n$

Published on 14 Oct 2020 - 16:33

score 61 · Answer 8 · 2020-10-15 03:00:23

61

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Summation involving unsigned Stirling numbers of the first kind: $\sum_{n=0}^{l}{{n+v \brack n+2}}$

Published on 15 Oct 2020 - 3:00

score 1.1k · Answer 9 · 2026-03-27 07:16:58

1.1k

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Why are Stirling numbers of the first kind related to the number of permutations with $k$ cycles?

Published on 27 Mar 2026 - 7:16

score 259 · Answer 10 · 2020-10-23 20:41:03

259

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Relationship between Stirling numbers of the first kind and their absolute value (possible error)

Published on 23 Oct 2020 - 20:41

score 70 · Answer 11 · 2026-03-28 06:59:59

70

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Software for evaluating summation expressions involving Stirling numbers of the first kind.

Published on 28 Mar 2026 - 6:59

score 278 · Answer 12 · 2020-10-29 16:24:47

278

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Closed-form for $ \sum_{n=0}^{\infty} \frac{x^n}{n!}n^p, $ where $p\in\mathbb{N}$

Published on 29 Oct 2020 - 16:24

score 129 · Answer 13 · 2020-11-09 20:49:36

129

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Proof of the Stirling Number Inequality, $S(m, n-2) \leq \binom{n}{3}S(m,n)$

Published on 09 Nov 2020 - 20:49

score 56 · Answer 14 · 2020-11-19 02:07:23

56

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Signless Stirling numbers of the first kind with minimum 2 elements

Published on 19 Nov 2020 - 2:07

score 543 · Answer 15 · 2020-11-21 17:06:02

543

Views

10 ball as put in 7 boxes randomly. What is the probability that at least one box is empty?

Published on 21 Nov 2020 - 17:06

List Question

Сombinatorial proof for recurrent formula of stirling's numbers

Give combinatorial proof for Stirling number of second kind formula: $S(n,k) = \sum_{i=1}^{n}S(n-i, k-2)(k^{i-1}-(k-1)^{i-1})$

Use combinatorics to prove $L\left(n,k\right)=\sum_{j=0}^{n}{ n\brack j}{j\brace k}$

Using combinatorial argument to prove that ${ n\brack 3}=\frac{1}{2}\left(n-1\right)!\left[\left(H_{n-1}\right)^{2}-H_{n-1}^{\left(2\right)}\right]$

A Stirling number identity

Number of ways to distribute balls in three out of five boxes

Prove that for all positive integers $n$, $\sum_{k=1}^{n}(-1)^kk!S(n,k)=(-1)^n$

Summation involving unsigned Stirling numbers of the first kind: $\sum_{n=0}^{l}{{n+v \brack n+2}}$

Why are Stirling numbers of the first kind related to the number of permutations with $k$ cycles?

Relationship between Stirling numbers of the first kind and their absolute value (possible error)

Software for evaluating summation expressions involving Stirling numbers of the first kind.

Closed-form for $ \sum_{n=0}^{\infty} \frac{x^n}{n!}n^p, $ where $p\in\mathbb{N}$

Proof of the Stirling Number Inequality, $S(m, n-2) \leq \binom{n}{3}S(m,n)$

Signless Stirling numbers of the first kind with minimum 2 elements

10 ball as put in 7 boxes randomly. What is the probability that at least one box is empty?

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