Question

Combinatorial proof of $S(n,n-2)=\binom{n}{3}+\frac{1}{2}\binom{n}{2}\binom{n-2}{2}$ when $n \ge 4$

score 82 · Answer 1 · 2026-03-27 17:04:21

82

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Combinatorial proof of $S(n,n-2)=\binom{n}{3}+\frac{1}{2}\binom{n}{2}\binom{n-2}{2}$ when $n \ge 4$

Published on 27 Mar 2026 - 17:04

score 145 · Answer 2 · 2023-03-01 15:32:43

145

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Show that player I can always win a Nim game in which the number of heaps with an odd number of coins is odd

Published on 01 Mar 2023 - 15:32

score 592 · Answer 3 · 2023-03-02 19:40:04

592

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Prove that if $2n$ points are colored red or blue, then we can always connect the red points to the blue points with non-intersecting line segments.

Published on 02 Mar 2023 - 19:40

score 57 · Answer 4 · 2026-03-25 22:11:15

57

Views

To prove that no perfect covers exist for the staircase board.

Published on 25 Mar 2026 - 22:11

score 159 · Answer 5 · 2026-03-27 17:04:29

159

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Inequality for triple product of Stirling numbers

Published on 27 Mar 2026 - 17:04

score 16 · Answer 6 · 2026-02-23 07:40:35

16

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"halfthink sets" contain n and n+1

Published on 23 Feb 2026 - 7:40

score 137 · Answer 7 · 2023-03-09 21:12:03

137

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Combinatorially proving $F_{n} = \sum_{i=0}^{\left\lfloor n/2 \right\rfloor}\binom{n-i}{i}$, where $F_n$ is the $n$-th Fibonacci number

Published on 09 Mar 2023 - 21:12

score 38 · Answer 8 · 2023-03-11 13:20:38

38

Views

Proof of recursive relationship between binary permutations.

Published on 11 Mar 2023 - 13:20

score 98 · Answer 9 · 2023-03-14 05:49:52

Prove combinatorially that: $\displaystyle {{n}\choose{k}} {{n}\choose{m}} = \sum^{k}_{i=0} {{n}\choose{m+i}}{{m+i}\choose{k}} {{k}\choose{i}}$

score 63 · Answer 10 · 2023-03-21 19:20:25

63

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${r + s \choose r + s - n} = \sum_{p,q,p',q'} {r \choose p} {s \choose q}$

Published on 21 Mar 2023 - 19:20

score 57 · Answer 11 · 2023-03-22 12:59:36

57

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Find all natural numbers $n$ such that $\binom{n+1}{k}$ is even $\forall~k=1,\dots,n$.

Published on 22 Mar 2023 - 12:59

score 329 · Answer 12 · 2026-03-25 17:33:50

329

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Let $S_1, S_2, \dots , S_m$ be distinct subsets of $\{1, 2, \dots , n\}$ such that $|S_i \cap S_j | = 1$ for all $i \ne j$. Prove that $m \le n$.

Published on 25 Mar 2026 - 17:33

score 137 · Answer 13 · 2023-03-29 14:09:22

137

Views

Find sum $S = {n \choose 1} - 2 {n \choose 2} + 3{n \choose 3} - 4{n \choose 4} + \cdots + (-1)^{n} n {n \choose n}$

Published on 29 Mar 2023 - 14:09

score 1.1k · Answer 14 · 2023-03-30 02:11:41

1.1k

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Counting the number of ways of organizing an odd number of objects into 3 odd groups, elegant method?

Published on 30 Mar 2023 - 2:11

score 91 · Answer 15 · 2023-04-03 01:08:21

How to construct a combinatorial proof of the identity $\sum_{t=0}^m {m \choose t} {2m \choose t} = {3m \choose m}$ by counting the number of paths

List Question

Combinatorial proof of $S(n,n-2)=\binom{n}{3}+\frac{1}{2}\binom{n}{2}\binom{n-2}{2}$ when $n \ge 4$

Show that player I can always win a Nim game in which the number of heaps with an odd number of coins is odd

Prove that if $2n$ points are colored red or blue, then we can always connect the red points to the blue points with non-intersecting line segments.

To prove that no perfect covers exist for the staircase board.

Inequality for triple product of Stirling numbers

"halfthink sets" contain n and n+1

Combinatorially proving $F_{n} = \sum_{i=0}^{\left\lfloor n/2 \right\rfloor}\binom{n-i}{i}$, where $F_n$ is the $n$-th Fibonacci number

Proof of recursive relationship between binary permutations.

Prove combinatorially that: $\displaystyle {{n}\choose{k}} {{n}\choose{m}} = \sum^{k}_{i=0} {{n}\choose{m+i}}{{m+i}\choose{k}} {{k}\choose{i}}$

${r + s \choose r + s - n} = \sum_{p,q,p',q'} {r \choose p} {s \choose q}$

Find all natural numbers $n$ such that $\binom{n+1}{k}$ is even $\forall~k=1,\dots,n$.

Let $S_1, S_2, \dots , S_m$ be distinct subsets of $\{1, 2, \dots , n\}$ such that $|S_i \cap S_j | = 1$ for all $i \ne j$. Prove that $m \le n$.

Find sum $S = {n \choose 1} - 2 {n \choose 2} + 3{n \choose 3} - 4{n \choose 4} + \cdots + (-1)^{n} n {n \choose n}$

Counting the number of ways of organizing an odd number of objects into 3 odd groups, elegant method?

How to construct a combinatorial proof of the identity $\sum_{t=0}^m {m \choose t} {2m \choose t} = {3m \choose m}$ by counting the number of paths

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