Question

Story proof for choosing a committee in a combinatorial identity: $\sum_{k=0}^n k{n\choose k}^2 = {n{2n-1\choose n-1}}$

score 1.6k · Answer 1 · 2014-11-25 19:10:51

1.6k

Views

Story proof for choosing a committee in a combinatorial identity: $\sum_{k=0}^n k{n\choose k}^2 = {n{2n-1\choose n-1}}$

Published on 25 Nov 2014 - 19:10

score 138 · Answer 2 · 2014-11-26 15:36:33

138

Views

Give an combinatorial argument

Published on 26 Nov 2014 - 15:36

score 61 · Answer 3 · 2014-11-27 01:18:21

61

Views

Question about some properties of combinatorial structures

Published on 27 Nov 2014 - 1:18

score 1.3k · Answer 4 · 2014-12-26 12:51:59

1.3k

Views

Combinatoric proof for $\sum_{k=0}^n{n\choose k}\left(-1\right)^k\left(n-k\right)^4 = 0$ ($n\geqslant5$)

Published on 26 Dec 2014 - 12:51

score 355 · Answer 5 · 2014-12-26 13:56:51

355

Views

Is there a "counting groups/committees" proof for the identity $\binom{\binom{n}{2}}{2}=3\binom{n+1}{4}$?

Published on 26 Dec 2014 - 13:56

score 266 · Answer 6 · 2019-10-02 20:32:11

266

Views

Bijective Proof: $\sum_{k=0}^n \binom{n}{k} (2k-1)!! (2n-2k-1)!!=2^n \cdot n!$

Published on 02 Oct 2019 - 20:32

score 52 · Answer 7 · 2019-10-11 07:03:36

52

Views

To prove that $a_n=[x^n]\frac {x-2x^2+x^3}{1-2x+x^2-x^4}$

Published on 11 Oct 2019 - 7:03

score 72 · Answer 8 · 2019-10-13 22:24:32

72

Views

Every person plays with another in a tournament of $n$ people.

Published on 13 Oct 2019 - 22:24

score 90 · Answer 9 · 2019-10-14 13:45:56

90

Views

How do I prove an equality by constructing a bijection

Published on 14 Oct 2019 - 13:45

score 266 · Answer 10 · 2026-03-25 09:27:20

266

Views

Algebraic proof of $\sum_{k}\binom{n}{2k}\binom{2k}{k}2^{n-2k}=\binom{2n}n$ (Combinatoric proof is given)

Published on 25 Mar 2026 - 9:27

score 769 · Answer 11 · 2019-10-23 20:06:03

769

Views

In how many ways can we choose 3 subsets, $A$, $B$ and $C$ From set $S$ with $n$ elements so that the union is $S$.

Published on 23 Oct 2019 - 20:06

score 260 · Answer 12 · 2019-10-24 22:49:27

260

Views

Combinatorial Proof of $\binom{n+1}{k} = (n+1)\binom{n}{k-1}/k$

Published on 24 Oct 2019 - 22:49

score 366 · Answer 13 · 2019-10-25 06:58:53

366

Views

To prove that the total number of symmetric partitions of $n$ is equal to $[x^n]\prod_{j=1}^{\infty}(1+x^{2j-1})$

Published on 25 Oct 2019 - 6:58

score 88 · Answer 14 · 2019-10-28 23:12:08

88

Views

A simple problem about edge bicolored complete graphs

Published on 28 Oct 2019 - 23:12

score 64 · Answer 15 · 2026-03-25 11:05:18

64

Views

Combinatorial proof of the identity relating Hurwitz numbers

Published on 25 Mar 2026 - 11:05

List Question

Story proof for choosing a committee in a combinatorial identity: $\sum_{k=0}^n k{n\choose k}^2 = {n{2n-1\choose n-1}}$

Give an combinatorial argument

Question about some properties of combinatorial structures

Combinatoric proof for $\sum_{k=0}^n{n\choose k}\left(-1\right)^k\left(n-k\right)^4 = 0$ ($n\geqslant5$)

Is there a "counting groups/committees" proof for the identity $\binom{\binom{n}{2}}{2}=3\binom{n+1}{4}$?

Bijective Proof: $\sum_{k=0}^n \binom{n}{k} (2k-1)!! (2n-2k-1)!!=2^n \cdot n!$

To prove that $a_n=[x^n]\frac {x-2x^2+x^3}{1-2x+x^2-x^4}$

Every person plays with another in a tournament of $n$ people.

How do I prove an equality by constructing a bijection

Algebraic proof of $\sum_{k}\binom{n}{2k}\binom{2k}{k}2^{n-2k}=\binom{2n}n$ (Combinatoric proof is given)

In how many ways can we choose 3 subsets, $A$, $B$ and $C$ From set $S$ with $n$ elements so that the union is $S$.

Combinatorial Proof of $\binom{n+1}{k} = (n+1)\binom{n}{k-1}/k$

To prove that the total number of symmetric partitions of $n$ is equal to $[x^n]\prod_{j=1}^{\infty}(1+x^{2j-1})$

A simple problem about edge bicolored complete graphs

Combinatorial proof of the identity relating Hurwitz numbers

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