Question

Counting argument for $\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}\binom{n}{j}\frac{1}{k} = \frac{1}{n}$

score 197 · Answer 1 · 2019-09-12 07:58:56

197

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Counting argument for $\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}\binom{n}{j}\frac{1}{k} = \frac{1}{n}$

Published on 12 Sep 2019 - 7:58

score 1.8k · Answer 2 · 2019-09-14 18:38:58

1.8k

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expanding a factorial by simplifying with $(k-1)!$

Published on 14 Sep 2019 - 18:38

score 200 · Answer 3 · 2019-09-16 20:40:14

200

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Derive a Combinatorial Identity

Published on 16 Sep 2019 - 20:40

score 291 · Answer 4 · 2019-09-22 20:02:43

291

Views

counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$

Published on 22 Sep 2019 - 20:02

score 170 · Answer 5 · 2019-09-24 23:45:03

170

Views

Number of zeroes and sign switches in a constructed zero-sum double sequence

Published on 24 Sep 2019 - 23:45

score 365 · Answer 6 · 2019-09-27 00:58:26

365

Views

proving the identity $b^3 = 6\binom{b}{3} + 6\binom{b}{2} + b$ for $b \in \mathbb{N}$, $b > 2$, by counting a set in different ways

Published on 27 Sep 2019 - 0:58

score 172 · Answer 7 · 2019-09-27 11:12:36

To prove for multichoose $\big(\!{{a+b}\choose k}\!\big)= \sum_{j=0}^k \big(\!{a\choose j}\!\big) \cdot \big(\!{b\choose {k-j}}\!\big)$

score 553 · Answer 8 · 2019-09-30 21:55:48

553

Views

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

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score 2.6k · Answer 9 · 2026-03-25 20:09:45

2.6k

Views

The properties of graph and its relation with the largest eigenvalue

Published on 25 Mar 2026 - 20:09

score 1.6k · Answer 10 · 2014-03-13 05:36:08

1.6k

Views

Prove $1^2+2^2+\cdots+n^2 = {n+1\choose2}+2{n+1\choose3}$

Published on 13 Mar 2014 - 5:36

score 80 · Answer 11 · 2014-03-14 21:37:38

80

Views

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score 404 · Answer 12 · 2014-04-06 15:49:49

404

Views

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Published on 06 Apr 2014 - 15:49

score 297 · Answer 13 · 2014-04-11 08:41:38

297

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Finite sum identity involving Stirling numbers

Published on 11 Apr 2014 - 8:41

score 23.3k · Answer 14 · 2014-04-20 23:38:30

23.3k

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Combinatorial explanation for why $n^2 = {n \choose 2} + {n+1 \choose 2}$

Published on 20 Apr 2014 - 23:38

score 402 · Answer 15 · 2014-05-15 07:13:24

402

Views

Combinatorial proof of $k\binom{n}{k} = n\binom{n-1}{k-1}$

Published on 15 May 2014 - 7:13

List Question

Counting argument for $\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}\binom{n}{j}\frac{1}{k} = \frac{1}{n}$

expanding a factorial by simplifying with $(k-1)!$

Derive a Combinatorial Identity

counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$

Number of zeroes and sign switches in a constructed zero-sum double sequence

proving the identity $b^3 = 6\binom{b}{3} + 6\binom{b}{2} + b$ for $b \in \mathbb{N}$, $b > 2$, by counting a set in different ways

To prove for multichoose $\big(\!{{a+b}\choose k}\!\big)= \sum_{j=0}^k \big(\!{a\choose j}\!\big) \cdot \big(\!{b\choose {k-j}}\!\big)$

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

The properties of graph and its relation with the largest eigenvalue

Prove $1^2+2^2+\cdots+n^2 = {n+1\choose2}+2{n+1\choose3}$

Confirming the correctness of Combinatoric permutation formula

Combinatorial proofs - how?

Finite sum identity involving Stirling numbers

Combinatorial explanation for why $n^2 = {n \choose 2} + {n+1 \choose 2}$

Combinatorial proof of $k\binom{n}{k} = n\binom{n-1}{k-1}$

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