Question

A combinatorial proof of the identity: $\sum_{k=1}^n k \binom{n}{k}^2 = {n}\binom{2n-1}{n-1}$?

score 2.7k · Answer 1 · 2017-12-11 03:29:08

2.7k

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A combinatorial proof of the identity: $\sum_{k=1}^n k \binom{n}{k}^2 = {n}\binom{2n-1}{n-1}$?

Published on 11 Dec 2017 - 3:29

score 446 · Answer 2 · 2026-03-25 16:09:40

446

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Combinatorial proof of the identity ${{n}\brack {m}}_{q}{{m}\brack {k}}_{q} = {{n}\brack {k}}_{q}{{n-k}\brack {m-k}}_{q}$

Published on 25 Mar 2026 - 16:09

score 832 · Answer 3 · 2026-03-25 03:07:18

832

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Bijection between perfect matchings permutations with even cycles

Published on 25 Mar 2026 - 3:07

score 179 · Answer 4 · 2017-12-23 15:41:14

179

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Combinatorial proof of $\sum_{k} \binom{2r}{2k-1}\binom{k-1}{s-1}=2^{2r-2s+1}\binom{2r-s}{s-1}\ \ r,s\in \mathbb{N}_0$

Published on 23 Dec 2017 - 15:41

score 1.8k · Answer 5 · 2017-12-27 11:27:38

1.8k

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Let $n$ be a positive integer, Prove that $\sum_{k=1}^n\frac{ (-1)^{k-1}}{k}{n \choose k} = H_n$

Published on 27 Dec 2017 - 11:27

score 382 · Answer 6 · 2017-12-30 02:39:18

382

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Can you elaborate the following combinatorial proof in details?

Published on 30 Dec 2017 - 2:39

score 235 · Answer 7 · 2017-12-30 04:28:24

235

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Combinatorial proof for $\sum_{r=0} {n \choose {2r}} = \sum_{r=0} {n \choose {2r+1}} = 2^{n-1}$

Published on 30 Dec 2017 - 4:28

score 111 · Answer 8 · 2017-12-30 07:54:48

111

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The combinatorial identity.

Published on 30 Dec 2017 - 7:54

score 345 · Answer 9 · 2018-01-01 11:19:50

345

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Is there an algebraic proof for $\sum_{m=k}^{n-k} \binom{m}{k}\binom{n-m}{k} = \binom{n+1}{2k+1}, n\ge2k\ge0$

Published on 01 Jan 2018 - 11:19

score 211 · Answer 10 · 2018-01-03 17:22:57

211

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A problem on combinatorics, boxes and dispositions

Published on 03 Jan 2018 - 17:22

score 208 · Answer 11 · 2026-02-23 11:19:59

208

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Bijective proof that $8+1=9$, or really $3^2-1=2^3$

Published on 23 Feb 2026 - 11:19

score 1k · Answer 12 · 2018-01-11 21:01:14

1k

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Proving with combinatorics $n^3=(n-1)^3+3(n-1)^2 +3(n-1)+1$

Published on 11 Jan 2018 - 21:01

score 3.4k · Answer 13 · 2010-11-23 04:22:22

3.4k

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Combinatorial proof of a binomial coefficient summation: $\sum_{k=1}^n \binom nk \binom n{k-1} = \frac12\binom{2n+2}{n+1} - \binom{2n}n$

Published on 23 Nov 2010 - 4:22

score 3.7k · Answer 14 · 2010-12-24 22:00:01

3.7k

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Combinatorial proof of a Fibonacci identity: $n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3.$

Published on 24 Dec 2010 - 22:00

score 3.2k · Answer 15 · 2010-12-25 16:30:34

3.2k

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Combinatorial Identity $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$

Published on 25 Dec 2010 - 16:30

List Question

A combinatorial proof of the identity: $\sum_{k=1}^n k \binom{n}{k}^2 = {n}\binom{2n-1}{n-1}$?

Combinatorial proof of the identity ${{n}\brack {m}}_{q}{{m}\brack {k}}_{q} = {{n}\brack {k}}_{q}{{n-k}\brack {m-k}}_{q}$

Bijection between perfect matchings permutations with even cycles

Combinatorial proof of $\sum_{k} \binom{2r}{2k-1}\binom{k-1}{s-1}=2^{2r-2s+1}\binom{2r-s}{s-1}\ \ r,s\in \mathbb{N}_0$

Let $n$ be a positive integer, Prove that $\sum_{k=1}^n\frac{ (-1)^{k-1}}{k}{n \choose k} = H_n$

Can you elaborate the following combinatorial proof in details?

Combinatorial proof for $\sum_{r=0} {n \choose {2r}} = \sum_{r=0} {n \choose {2r+1}} = 2^{n-1}$

The combinatorial identity.

Is there an algebraic proof for $\sum_{m=k}^{n-k} \binom{m}{k}\binom{n-m}{k} = \binom{n+1}{2k+1}, n\ge2k\ge0$

A problem on combinatorics, boxes and dispositions

Bijective proof that $8+1=9$, or really $3^2-1=2^3$

Proving with combinatorics $n^3=(n-1)^3+3(n-1)^2 +3(n-1)+1$

Combinatorial proof of a binomial coefficient summation: $\sum_{k=1}^n \binom nk \binom n{k-1} = \frac12\binom{2n+2}{n+1} - \binom{2n}n$

Combinatorial proof of a Fibonacci identity: $n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3.$

Combinatorial Identity $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$

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