Question

Use combinatorial reasoning to show that Stirling number

score 90 · Answer 1 · 2020-05-10 18:50:42

90

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Use combinatorial reasoning to show that Stirling number

Published on 10 May 2020 - 18:50

score 24 · Answer 2 · 2020-05-13 20:45:24

24

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How do I derive the formula S(G,k)=S(G+e,k)+S(G/e,k)?

Published on 13 May 2020 - 20:45

score 143 · Answer 3 · 2020-05-21 10:55:06

143

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Combinatorial proof of ${n\brace k}= \frac{k^{n}}{k!}-\sum_{r=1}^{k-1}\frac{ {n\brace r}}{\left(k-r\right)!}$

Published on 21 May 2020 - 10:55

score 106 · Answer 4 · 2020-05-22 07:16:22

106

Views

Proof of $\sum_{n\ge0}^{ }n^{m}z^{n}=\sum_{j=0}^{m}{m+1\brace j+1}\frac{\left(-1\right)^{m-j}j!}{\left(1-z\right)^{j+1}}$

Published on 22 May 2020 - 7:16

score 520 · Answer 5 · 2020-05-22 08:15:11

520

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Range of $m,n$ in $\sum_{j=0}^{m}{m\brace j}\binom{n}{j}j!=n^{m}$

Published on 22 May 2020 - 8:15

score 168 · Answer 6 · 2026-03-25 11:10:47

168

Views

An approximation of the ordered Bell numbers

Published on 25 Mar 2026 - 11:10

score 117 · Answer 7 · 2020-06-30 21:40:03

117

Views

Proof of identity relating Prime-counting function and Stirling numbers of the second kind

Published on 30 Jun 2020 - 21:40

score 55 · Answer 8 · 2020-07-05 09:18:42

55

Views

Closed-form solution to the recurrence

Published on 05 Jul 2020 - 9:18

score 314 · Answer 9 · 2020-07-10 19:46:46

314

Views

Help in simplifying combinatorial sum $\frac{n!}{(n-k)!}-{1\over(n-k)!}{\sum _{m=1}^{k-1} (-1)^{m+1} (n-m)! S(k,k-m)}$

Published on 10 Jul 2020 - 19:46

score 133 · Answer 10 · 2020-07-15 00:46:17

133

Views

Number of ways of assigning n objects to k people such that each person has at least r?

Published on 15 Jul 2020 - 0:46

score 109 · Answer 11 · 2020-07-15 19:42:43

109

Views

Evaluating the recurrence $f_k(x)= f'_{k-1}(x)+f_{k-1}(x) f_1(x)$ with $f_0(x)=1$, $f_1(x)=e^x$

Published on 15 Jul 2020 - 19:42

score 192 · Answer 12 · 2020-07-21 17:40:01

192

Views

Find two-index Inverse relations

Published on 21 Jul 2020 - 17:40

score 226 · Answer 13 · 2026-03-25 12:47:21

226

Views

Exponential generating function with Stirling numbers

Published on 25 Mar 2026 - 12:47

score 2k · Answer 14 · 2016-07-06 20:40:00

2k

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stirling numbers of second kind

Published on 06 Jul 2016 - 20:40

score 952 · Answer 15 · 2016-07-27 21:44:38

952

Views

Sum identity using Stirling numbers of the second kind

Published on 27 Jul 2016 - 21:44

List Question

Use combinatorial reasoning to show that Stirling number

How do I derive the formula S(G,k)=S(G+e,k)+S(G/e,k)?

Combinatorial proof of ${n\brace k}= \frac{k^{n}}{k!}-\sum_{r=1}^{k-1}\frac{ {n\brace r}}{\left(k-r\right)!}$

Proof of $\sum_{n\ge0}^{ }n^{m}z^{n}=\sum_{j=0}^{m}{m+1\brace j+1}\frac{\left(-1\right)^{m-j}j!}{\left(1-z\right)^{j+1}}$

Range of $m,n$ in $\sum_{j=0}^{m}{m\brace j}\binom{n}{j}j!=n^{m}$

An approximation of the ordered Bell numbers

Proof of identity relating Prime-counting function and Stirling numbers of the second kind

Closed-form solution to the recurrence

Help in simplifying combinatorial sum $\frac{n!}{(n-k)!}-{1\over(n-k)!}{\sum _{m=1}^{k-1} (-1)^{m+1} (n-m)! S(k,k-m)}$

Number of ways of assigning n objects to k people such that each person has at least r?

Evaluating the recurrence $f_k(x)= f'_{k-1}(x)+f_{k-1}(x) f_1(x)$ with $f_0(x)=1$, $f_1(x)=e^x$

Find two-index Inverse relations

Exponential generating function with Stirling numbers

stirling numbers of second kind

Sum identity using Stirling numbers of the second kind

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