Question

Prove the identity ${}_{n}{\rm H}_{n-1}=\sum_{k=0}^{n-1} \left({}_{n-k}{\rm H}_{k} \right)^2 $ combinatorially

score 84 · Answer 1 · 2019-10-29 09:28:07

84

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Prove the identity ${}_{n}{\rm H}_{n-1}=\sum_{k=0}^{n-1} \left({}_{n-k}{\rm H}_{k} \right)^2 $ combinatorially

Published on 29 Oct 2019 - 9:28

score 107 · Answer 2 · 2019-10-29 15:08:39

107

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A problem about edge bicolored complete graphs 2

Published on 29 Oct 2019 - 15:08

score 73 · Answer 3 · 2019-10-29 16:36:57

73

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Combinations company

Published on 29 Oct 2019 - 16:36

score 263 · Answer 4 · 2019-10-29 18:41:09

263

Views

$99 \times 99$ colored square grid

Published on 29 Oct 2019 - 18:41

score 68 · Answer 5 · 2019-10-31 15:30:03

68

Views

Combinatorial Proof: $ {a \choose b} {b \choose c} = {a \choose a-c}{a-c \choose b-c}$

Published on 31 Oct 2019 - 15:30

score 92 · Answer 6 · 2019-11-08 03:33:04

92

Views

interpretation on $1^2+2^2+\ldots+n^2=\binom{n+1}{2}+2\binom{n+1}{3}$ and $1^3+2^3+\ldots+n^3=\binom{n+1}{2}+6\binom{n+1}{3}+6\binom{n+1}{4}$

Published on 08 Nov 2019 - 3:33

score 40 · Answer 7 · 2019-11-09 03:09:00

40

Views

Equivalence of two methods for possible outomes of $4$ dice problem

Published on 09 Nov 2019 - 3:09

score 169 · Answer 8 · 2019-11-12 08:56:50

169

Views

Combinatorial view of binomial identity

Published on 12 Nov 2019 - 8:56

score 37 · Answer 9 · 2019-11-12 12:58:03

37

Views

Why this solution does not works for the problem "Find number of ways to put 5 identical balls in two different bags"?

Published on 12 Nov 2019 - 12:58

score 107 · Answer 10 · 2019-11-15 06:02:46

107

Views

Show that for $r,n \in \mathbb{N}$ with $r \geq n$, $\sum_{i=0}^{n-1} (-1)^i \binom{n}{i} \binom{r+n-i-1}{r} = \binom{r-1}{n-1}$

Published on 15 Nov 2019 - 6:02

score 64 · Answer 11 · 2019-11-17 00:22:02

64

Views

stirling number of the first kind

Published on 17 Nov 2019 - 0:22

score 242 · Answer 12 · 2019-11-19 07:50:21

242

Views

finite sum with combinatorics

Published on 19 Nov 2019 - 7:50

score 133 · Answer 13 · 2019-11-20 21:20:32

133

Views

Number of permutations with a given property

Published on 20 Nov 2019 - 21:20

score 125 · Answer 14 · 2019-11-21 00:57:58

125

Views

Combinatorial proof for identity $\sum_{k=0}^{2n}(-1)^k\binom{3n}{k}=\binom{3n-1}{2n}$

Published on 21 Nov 2019 - 0:57

score 53 · Answer 15 · 2019-11-21 20:48:46

53

Views

Is there a combinatorial proof of the Taylor series $\frac1{\sqrt{1-4x}}=\sum_{n=0}^\infty{2n\choose n}x^n$?

Published on 21 Nov 2019 - 20:48

List Question

Prove the identity ${}_{n}{\rm H}_{n-1}=\sum_{k=0}^{n-1} \left({}_{n-k}{\rm H}_{k} \right)^2 $ combinatorially

A problem about edge bicolored complete graphs 2

Combinations company

$99 \times 99$ colored square grid

Combinatorial Proof: $ {a \choose b} {b \choose c} = {a \choose a-c}{a-c \choose b-c}$

interpretation on $1^2+2^2+\ldots+n^2=\binom{n+1}{2}+2\binom{n+1}{3}$ and $1^3+2^3+\ldots+n^3=\binom{n+1}{2}+6\binom{n+1}{3}+6\binom{n+1}{4}$

Equivalence of two methods for possible outomes of $4$ dice problem

Combinatorial view of binomial identity

Why this solution does not works for the problem "Find number of ways to put 5 identical balls in two different bags"?

Show that for $r,n \in \mathbb{N}$ with $r \geq n$, $\sum_{i=0}^{n-1} (-1)^i \binom{n}{i} \binom{r+n-i-1}{r} = \binom{r-1}{n-1}$

stirling number of the first kind

finite sum with combinatorics

Number of permutations with a given property

Combinatorial proof for identity $\sum_{k=0}^{2n}(-1)^k\binom{3n}{k}=\binom{3n-1}{2n}$

Is there a combinatorial proof of the Taylor series $\frac1{\sqrt{1-4x}}=\sum_{n=0}^\infty{2n\choose n}x^n$?

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