List questions from combinatorial-proofs

Published on 09 Mar 2022 - 21:09

161

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Number of Subsets of Cardinality at most $k$ of a Finite Set

Published on 09 Mar 2022 - 22:38

87

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Proving the identity $\sum_{k = r}^n {n \choose k}{k \choose r}2^k = {n \choose r}2^r3^{n - 2}$

Published on 10 Mar 2022 - 12:54

106

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binomial identity: $\sum_{k=x+y}^{\infty}\binom{k-1}{y-1}\binom{k-y}{x}u^k = \binom{x+y-1}{y-1}\left(\frac{u}{1-u}\right)^{x+y}$?

Published on 11 Mar 2022 - 21:18

57

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Combinatorial argument for this problem:

Published on 12 Mar 2022 - 3:04

112

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Proving the identity $\sum_{k = 1}^nk^2 {n \choose k}^2 = n^2{2n - 2 \choose n - 1}$ combinatorially

Published on 12 Mar 2022 - 13:54

598

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$k$-permutations combinatorial proof $P(n,k) = P(n-1,k) + k\cdot P(n-1,k-1).$

Published on 14 Mar 2022 - 4:39

390

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Proof of $(n)_k = (n-1)_k + k(n-1)_{k-1}$

Published on 14 Mar 2022 - 16:57

50

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Injection from tuples of subsets to others

Published on 16 Mar 2022 - 23:00

298

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XOR operation inner-product proof

Published on 25 Mar 2026 - 9:31

135

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Finding sum of Stirling numbers (signed) of the first kind $\sum_{k=0}^n s(n,k)$

Published on 30 Mar 2026 - 5:13

143

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Number of distinct derivatives of order $n$ of an $m$-variable function

Published on 27 Mar 2022 - 19:47

62

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A combinatorial problem of counting path weights with a special symbolic binary tree

Published on 30 Mar 2022 - 4:37

155

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Combinatorial interpretation of a sum

Published on 31 Mar 2022 - 0:25

150

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Combinatorial argument why is the following below true?

Published on 31 Mar 2022 - 9:28

List Question

Integer partition generating functions problem

Number of Subsets of Cardinality at most $k$ of a Finite Set

Proving the identity $\sum_{k = r}^n {n \choose k}{k \choose r}2^k = {n \choose r}2^r3^{n - 2}$

binomial identity: $\sum_{k=x+y}^{\infty}\binom{k-1}{y-1}\binom{k-y}{x}u^k = \binom{x+y-1}{y-1}\left(\frac{u}{1-u}\right)^{x+y}$?

Combinatorial argument for this problem:

Proving the identity $\sum_{k = 1}^nk^2 {n \choose k}^2 = n^2{2n - 2 \choose n - 1}$ combinatorially

$k$-permutations combinatorial proof $P(n,k) = P(n-1,k) + k\cdot P(n-1,k-1).$

Proof of $(n)_k = (n-1)_k + k(n-1)_{k-1}$

Injection from tuples of subsets to others

XOR operation inner-product proof

Finding sum of Stirling numbers (signed) of the first kind $\sum_{k=0}^n s(n,k)$

Number of distinct derivatives of order $n$ of an $m$-variable function

A combinatorial problem of counting path weights with a special symbolic binary tree

Combinatorial interpretation of a sum

Combinatorial argument why is the following below true?

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