Questions tagged [bijective-combinatorics]

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39 votes
7 answers
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Bijection between irreducible representations and conjugacy classes of finite groups

Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S_n$)?
Dan's user avatar
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17 votes
3 answers
707 views

Automated search for bijective proofs

In enumerative combinatorics, a bijective proof that $|A_n| = |B_n|$ (where $A_n$ and $B_n$ are finite sets of combinatorial objects of size $n$) is a proof that constructs an explicit bijection ...
Timothy Chow's user avatar
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10 votes
1 answer
715 views

Curious Catalan convolutions

Question. Do these identities involving even-index Catalan numbers have a known combinatorial interpretation? They look as though they should. I haven’t seen one in the literature. $$\sum_{a+b=n}C_{...
Robin Houston's user avatar
8 votes
3 answers
3k views

Permutations with all cycles odd length and permutations with all cycles even length

If $n$ is even, then the number of permutations of $n$ in which all cycles have odd length equals the number of permutations of $n$ in which all cycles have even length. This fact is easily proved, ...
Timothy Chow's user avatar
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12 votes
0 answers
318 views

Bijective proof of an identity involving number of standard Young tableaux and semistandard tableaux

Question. Can you find a bijective proof of the identity $$ \operatorname{dim}(S^{\lambda} \mathbb{C}^m)\ \operatorname{dim}(S^{\lambda'} \mathbb{C}^n) \ f^{n^m} = \dim \Lambda^p (\mathbb{C}^m \...
Piotr Śniady's user avatar
10 votes
0 answers
342 views

A bijective proof for the odd companion to Shapiro's Catalan convolution

Shapiro's Catalan convolution is the following formula (where $C_n$ is the $n$th Catalan number): $$ \sum_{k=0}^{n}{C_{2k}C_{2(n-k)}}=4^nC_n. $$ In other words, letting $C(z)=\sum_{n=0}^{\infty}{C_nz^...
Alexander Burstein's user avatar
8 votes
3 answers
1k views

Bijective proof for a partition identity

I came across the following cute fact about partitions: \begin{align} & |\{\lambda \vdash n \text{ with an even number of even parts}\}| \\[8pt] & {} - |\{ \lambda \vdash n \text{ with an odd ...
Nate's user avatar
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