# A bijection between sets of Young tableaux of two kinds

Assuming that the problem of exhibiting a bijection is not considerd a frivolous pursuit, allow me to ask a question troubling me for some time now.

Let $\lambda \vdash n$ denote the fact that $\lambda$ is a partition of $n$. Denote the number of parts by $l(\lambda)$. If $T$ is a standard Young tableau (SYT), we will denote the underlying partition shape by $sh(T)$.

Given a positive even integer $2n$, let $$Pe_{2n}=\{ \lambda: \lambda\vdash 2n,\text{ } l(\lambda) \leq3 \text{ and all parts of } \lambda \text{ are even} \}$$ and $$Qe_{2n}=\{ \lambda: \lambda\vdash 2n, \lambda = (k,k,1^{2n-2k}), \text{ }k\geq 1 \}$$

Using these sets we will define two more sets whose elements are SYTs. $$TPe_{2n}=\{T: T \text{ an SYT, } sh(T)\in Pe_{2n} \}$$ and $$TQe_{2n}=\{T: T \text{ an SYT, } sh(T)\in Qe_{2n} \}$$

$\textbf{Question}$: Is there a bijective proof exhibiting the fact that the cardinalities of $TPe_{2n}$ and $TQe_{2n}$ are equal?

The second question is very similar. Given an odd positive integer $2n+1$, let $$Po_{2n+1}=\{ \lambda: \lambda\vdash 2n+1,\text{ } l(\lambda)=3 \text{ and all parts of } \lambda \text{ are odd} \}$$ and $$Qo_{2n+1}=\{ \lambda: \lambda\vdash 2n+1, \lambda = (k,k,1^{2n+1-2k}), \text{ }k\geq 1 \}$$

Using these sets we will define two more sets whose elements are SYTs. $$TPo_{2n+1}=\{T: T \text{ an SYT, } sh(T)\in Po_{2n+1} \}$$ and $$TQo_{2n+1}=\{T: T \text{ an SYT, } sh(T)\in Qo_{2n+1} \}$$

$\textbf{Question}$: Is there a bijective proof exhibiting the fact that the cardinalities of $TPo_{2n+1}$ and $TQo_{2n+1}$ are equal?

I tried quite a few approaches ( Motzkin path interpretations, matching diagrams etc) but did not succeed. I hope somebody here can guide me.

The relevant OEIS entry would be link text

Thanks,

Vasu

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Just out of curiosity, how did you discover these identities? What is the proof you currently have? And lastly, have you looked into A.Regev's most recent papers (on arXiv)? There might be a bijection using the number of peaks of Motzkin paths... – Gjergji Zaimi Jul 14 '11 at 1:01