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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
    
@Gjergji: I was working on the computation of Kronecker coefficients and was intrigued by certain identities relating to counting tableaux of bounded height implied by them. You might want to look into work by Garsia-Xin-Zabrocki on computing $s_{(n,n)}*s_{(n,n)} and other examples.(one of the identities I happened to find was listed in this question: mathoverflow.net/questions/47070/…) –  Vasu vineet Jul 14 '11 at 3:14
    
... So I started playing around by counting Young tableaux with addition constraints and this is what I ended up with. One potential way to calculate $TPe_{2n}$ where I am actually counting fixed point free involutions with length of longest longest increasing subsequence bounded by 3, (I am sorry I haven't really worked out the details, and this was long back) would be to use Gessel's determinantal formulae as listed on Pg. 15 of Stanley's article on increasing and decreasing sequences and their variants (link: www-math.mit.edu/~rstan/papers/ids.pdf). –  Vasu vineet Jul 14 '11 at 3:14
    
... And to answer your last question, I have had a look at Regev's stuff but haven't managed to get anything concrete out of it. Since I am pretty certain the problem would yield to other approaches, I am very interested in a bijective proof –  Vasu vineet Jul 14 '11 at 3:17
    
I'm getting different sizes for $TP_{10}$ and $TQ_{10}$. Are you sure you don't have a typo somewhere? –  Gjergji Zaimi Sep 3 '11 at 17:23
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