6
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Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape $\lambda$, define the "flattened tableaux" by deleting the first row $\lambda_1$, and then relabeling all entries in the tableaux with respect to their relative order, thus giving a SYT $T'$ of shape $\lambda':=(\lambda_2,\cdots,\lambda_n)$ and $|\lambda'|=n-\lambda_1$. For example, if $\lambda=(4,3,2,1,1)$ with $|\lambda|=10$, then $\lambda'=(3,2,1,1)$. An as an example, take

$T=$ \begin{array}{cccc} 1 & 3 & 5 & 6\\ 2 & 4 & 7 & \ \\ 8 & 11 & \ & \ \\ 9 & \ & \ & \ \\ 10 & \ & \ &\ \end{array}

Then,

$T'=$ \begin{array}{cccc} 1 & 2 & 3 & \ \\ 4 & 7 & \ & \ \\ 5 & \ & \ & \ \\ 6 & \ & \ &\ \end{array}

Does this operation have a common name in literature? Has it been studied before? As a map, the flattening operation $\phi: SYT(\lambda)\rightarrow SYT(\lambda')$ is clearly a surjection (and not a bijection). On the other hand, are there specific $\lambda$ for which $\phi$ is uniform in $SYT(T')$? In other words, $|\{T: \ \phi(T)=T'\}|$ is the same for all $T'$?

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  • $\begingroup$ Why is this map an injection? In your example, swapping 6 with 7 gives the same SYT after flattening. $\endgroup$ Dec 2, 2014 at 9:26
  • $\begingroup$ @siddharth Venkatesh: meant to say surjection, pardon! $\endgroup$
    – Alex R.
    Dec 2, 2014 at 15:34

1 Answer 1

1
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For any partition $(n)$ or $(k,1^{n-k})$ , the resulting tableaux are respectively empty or a vertical one-column tableau with entries $1,2,...,n-k$. But for any partition $(k,2,1,1,..,1)$ with $n-k-2$ '$1$'s, the beheaded tableau can be either
$((1,2),(3),..,(n-k-1))$ or $((1,3),(2),..,(n-k-1))$
depending on $T$. So, beyond these special partitions $(n)$ and $(k,1^n-k)$ there can be no such cases.

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  • $\begingroup$ I don't understand the solution $\endgroup$
    – ArB
    Aug 30, 2021 at 17:20
  • $\begingroup$ I think he's answering the last question of OP with "no" by showing $T'$ with differing numbers of pre-images. In any case, "beheaded tableau" is great! $\endgroup$ Aug 31, 2021 at 0:13

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