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Results tagged with co.combinatorics
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user 6104
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
15
votes
2
answers
1k
views
A combinatorial identity involving harmonic numbers
The harmonic numbers are given by $$H_n=\sum_{k=1}^n\frac{1}{k}.$$
Numerical calculation suggests
$$
\sum_{k=1}^{n}(-1)^k{n\choose k}{n+k\choose k}\sum_{i=1}^{k}\frac{1}{n+i}=(-1)^nH_n.
$$
I can not g …
- 2,183
20
votes
1
answer
2k
views
How to prove that the following double sum is always an integer？
I have verified the following double sum is always an integer for $s$ up to $1000$ via Maple.
But I can not prove it. Proofs, hints, or references are all welcome.
Thanks!
$$\sum_{m=s}^{2s}\sum_{k=0}^{ …
- 2,183
21
votes
1
answer
871
views
A combinatorial identity involving generalized harmonic numbers
The $n$-th harmonic number is defined as
$$
H_n=\sum_{k=1}^{n}\frac{1}{k},
$$
and the generalized harmonic numbers are defined by
$$
H_{n}^{(r)}=\sum_{k=1}^{n}\frac{1}{k^r}.
$$
Recently, I have found …
- 2,183
47
votes
1
answer
4k
views
How to prove this polynomial always has integer values at all integers?
Let $m$ be any positive integer.
$$
P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}.
$$
Question: $P_m(x)$ always h …
- 2,183