Questions tagged [bijective-combinatorics]
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7 questions
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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 $S_n$)?
- 1,318
18
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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 ...
- 82.7k
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answer
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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.
$$\sum_{a+b=n}C_{...
- 2,896
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Permutations with all cycles odd length and permutations with all cycles even length
If $n$ is even, then the number of permutations of $n$ in which all cycles have odd length equals the number of permutations of $n$ in which all cycles have even length. This fact is easily proved, ...
- 82.7k
12
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Bijective proof of an identity involving number of standard Young tableaux and semistandard tableaux
Question. Can you find a bijective proof of the identity
$$ \operatorname{dim}(S^{\lambda} \mathbb{C}^m)\ \operatorname{dim}(S^{\lambda'} \mathbb{C}^n) \ f^{n^m}
= \dim \Lambda^p (\mathbb{C}^m \...
- 381
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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^...
- 1,190
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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 ...
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