The "lattice" in the title appears to be a lattice. At least it's a poset, which I define now.

Fix a partition $\lambda$ of $n$ and consider the set of all standard Young tableaux (each of $1,\dots,n$ occurs once in the tableau, and rows and columns are increasing) with shape $\lambda$. We define the poset by saying that $T'$ covers $T$ if $T'$ can be obtained by swapping the positions of entries $i$ and $i+1$ in $T$ for some $i$, such that $i$ moved to a strictly lower row.

Thus the $T'$:s covering a fixed $T$ are those given by swapping $i,i+1$ not in the same row or column such that $i$ is in a higher row than $i+1$.

I drew this poset for a few $\lambda$ and observed that it was a lattice.


1 Answer 1


Your poset is either the same or closely related to the one in http://www.fh-bingen.de/fileadmin/user_upload/Lehrende/Winkel_Rudolf/06.pdf, which is a lattice.

       Example 1.8
      (Figure added by J.O'Rourke.)

  • 1
    $\begingroup$ @Joseph O'Rourke, Thanks for the picture. $\endgroup$ Jul 5, 2013 at 0:43
  • $\begingroup$ Indeed it's my poset! Thanks for your answer. Apparently it also appears in a paper by Björner and Wachs. $\endgroup$
    – Erik
    Jul 5, 2013 at 19:54
  • $\begingroup$ You are welcome $\endgroup$ Jul 5, 2013 at 20:21

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