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As a modest start, I propose the below problem for a special set of partitions. Perhaps it is known.

Let $\lambda_n=(n,n-1,\dots,2,1)$ be the staircase partition and its corresponding Young diagram $Y_n$.

QUESTION. In how many different ways (denote this by $a_n$) can one tile $Y_n$ using monomers ($1\times1$ squares) and dimers ($1\times2$ or $2\times1$ rectangles)? Is there a determinant (Pfaffian) formulation of this enumeration, in Kasteleyn's style?

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    $\begingroup$ The monomer-dimer problem for rectangles is open. I would expect staircases to be of comparable difficulty. $\endgroup$
    – Richard Stanley
    Commented Apr 5, 2019 at 15:52
  • $\begingroup$ Thank you for your valuable feed-back, there is a new variation of this problem at mathoverflow.net/questions/327329/… $\endgroup$
    – T. Amdeberhan
    Commented Apr 6, 2019 at 17:02
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    $\begingroup$ This is now findstat.org/St001380 $\endgroup$
    – Martin Rubey
    Commented Apr 6, 2019 at 20:18

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