Question

Combinatorial proof verification: $\sum_{i=0}^n {{n}\choose{i}} { n-i\brack k} i! = { n+1 \brack k+1}$

score 63 · Answer 1 · 2023-08-08 05:04:21

63

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Combinatorial proof verification: $\sum_{i=0}^n {{n}\choose{i}} { n-i\brack k} i! = { n+1 \brack k+1}$

Published on 08 Aug 2023 - 5:04

score 56 · Answer 2 · 2023-08-08 17:26:04

56

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Combinatorial proof: $\sum_{k=1}^n k^m = \sum_{i=1}^m {{n+1}\choose{i+1}} \ {m\brace i} \ i!$. I don't understand one part of it (I made it wrong).

Published on 08 Aug 2023 - 17:26

score 81 · Answer 3 · 2023-08-09 04:38:23

81

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Combinatorial proof $\sum_{k=0}^n {{n}\choose{k}}^2 \ m^k = \sum_{j=0}^n {{n}\choose{j}} \ {{2n - j}\choose{n}} \ (m-1)^j$

Published on 09 Aug 2023 - 4:38

score 91 · Answer 4 · 2023-08-09 16:18:46

91

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Combinatorial proof for $n > k > 0$: $\sum_{j=0}^n {{k}\choose{j}} {{n-k}\choose{j}} = {{n}\choose{k}}$

Published on 09 Aug 2023 - 16:18

score 74 · Answer 5 · 2023-08-09 20:26:40

74

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Combinatorial proof for $n \in \mathbb{N}$: $\sum_{k=0}^n \sum_{j=0}^n {{k}\choose{j}} {{n-k}\choose{j}} = \sum_{m=0}^n {{n+1}\choose{2m+1}} $

Published on 09 Aug 2023 - 20:26

score 92 · Answer 6 · 2023-08-22 13:21:44

92

Views

Combinatorial proof for an integral

Published on 22 Aug 2023 - 13:21

score 31 · Answer 7 · 2023-08-26 07:22:10

31

Views

Combinations as generalized sequences

Published on 26 Aug 2023 - 7:22

score 78 · Answer 8 · 2023-08-27 13:58:24

78

Views

NIST Handbook Mathematical Functions: what is the binomial coefficient involving negative elements? what is the terminology for the rising factorial?

Published on 27 Aug 2023 - 13:58

score 234 · Answer 9 · 2023-08-30 18:34:48

234

Views

Proof that the two sums are equal - the glass ball problem.

Published on 30 Aug 2023 - 18:34

score 32 · Answer 10 · 2023-09-06 16:17:22

32

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How many solution for the inequality $n_1\varepsilon+n_2 \leq T$?

Published on 06 Sep 2023 - 16:17

score 60 · Answer 11 · 2026-03-26 03:00:25

60

Views

Fulton Harris Lemma A.28

Published on 26 Mar 2026 - 3:00

score 131 · Answer 12 · 2023-09-09 18:27:34

131

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Exercise 2 Stanley combinatorics volume 1: Bijective proof of a sum equality

Published on 09 Sep 2023 - 18:27

score 107 · Answer 13 · 2023-09-12 14:59:08

107

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Double summation manipulation

Published on 12 Sep 2023 - 14:59

score 29 · Answer 14 · 2023-09-19 14:42:53

29

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Prove that the product of $2n$ consecutive negative integers is divisible by $(2n)!$

Published on 19 Sep 2023 - 14:42

score 44 · Answer 15 · 2023-09-22 00:00:31

44

Views

Selecting r objects out of n, in order, proof?

Published on 22 Sep 2023 - 0:00

List Question

Combinatorial proof verification: $\sum_{i=0}^n {{n}\choose{i}} { n-i\brack k} i! = { n+1 \brack k+1}$

Combinatorial proof: $\sum_{k=1}^n k^m = \sum_{i=1}^m {{n+1}\choose{i+1}} \ {m\brace i} \ i!$. I don't understand one part of it (I made it wrong).

Combinatorial proof $\sum_{k=0}^n {{n}\choose{k}}^2 \ m^k = \sum_{j=0}^n {{n}\choose{j}} \ {{2n - j}\choose{n}} \ (m-1)^j$

Combinatorial proof for $n > k > 0$: $\sum_{j=0}^n {{k}\choose{j}} {{n-k}\choose{j}} = {{n}\choose{k}}$

Combinatorial proof for $n \in \mathbb{N}$: $\sum_{k=0}^n \sum_{j=0}^n {{k}\choose{j}} {{n-k}\choose{j}} = \sum_{m=0}^n {{n+1}\choose{2m+1}} $

Combinatorial proof for an integral

Combinations as generalized sequences

NIST Handbook Mathematical Functions: what is the binomial coefficient involving negative elements? what is the terminology for the rising factorial?

Proof that the two sums are equal - the glass ball problem.

How many solution for the inequality $n_1\varepsilon+n_2 \leq T$?

Fulton Harris Lemma A.28

Exercise 2 Stanley combinatorics volume 1: Bijective proof of a sum equality

Double summation manipulation

Prove that the product of $2n$ consecutive negative integers is divisible by $(2n)!$

Selecting r objects out of n, in order, proof?

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