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There is a plethora of polynomials defined on partition shaped Young diagrams, (Schur, Jack, Grothendieck,...), and skew Young diagrams. There are also composition shaped diagrams that are responsible for Demazure characters and some other non-symmetric polynomials, such as the non-symmetric Macdonald polynomials.

However, I am not aware of any family of polynomials that are defined on more general shapes of diagrams (well, except perhaps the shifted tableaux, that define $P$-Schur polynomials).

Thus, what are some families of polynomials (or families of fillings), defined on more exotically shaped diagrams?

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  • $\begingroup$ rook polynomials? $\endgroup$ May 5, 2015 at 20:03
  • $\begingroup$ Yeah, I was thinking about those, but most sources I have found only deal with Ferres shapes. Is there a source with more general arrangements? $\endgroup$ May 5, 2015 at 20:28
  • $\begingroup$ I'm afraid, I don't know. Sorry. $\endgroup$ May 6, 2015 at 6:32

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To any poset $P$ and any labeling $\omega$ of $P$ we can associate the ``polynomial'' (actually, formal power series) $K(P,\omega) := \sum_{\sigma \in A^r(P,\omega)} x^{\sigma}$. Here $A^r(P,\omega)$ is the set of all reverse $(P,\omega)$-partitions (i.e., fillings of the poset $P$ with entries $\mathbb{N}$ obeying certain inequalities depending on $\omega$ and the order structure) and we use the notation $x^f := \prod_{i \geq 1} x_i^{\#f^{-1}(i)}$. Then $K(P,\omega)$ is always a quasisymmetric function. See section 7.19 of Stanley's Enumerative Combinatorics vol 2 for more details.

When $P$ is the poset of a skew diagram and $\omega$ is a compatible ``Schur labelling'' we can say more: in this case $K_{(P,\omega)}$ is symmetric. In fact, I believe it is a conjecture of Stanley from circa 1970 that $K_{(P,\omega)}$ is a symmetric function if and only if $(P,\omega)$ is isomorphic to $(P_{\lambda/\mu},w)$, where $\lambda/\mu$ is a skew-shape and $w$ is a Schur labelling of $P_{\lambda/\mu}$.

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  • $\begingroup$ If you want to read more about that conjecture of Stanley's, see Malvenuto, Claudia. "P-partitions and the plactic congruence." Graphs and Combinatorics 9.1 (1993): 63-73. available at: wwwusers.di.uniroma1.it/~claudia/P-Partitions1.pdf $\endgroup$ May 5, 2015 at 18:38
  • $\begingroup$ Ah, that's a very interesting reference! $\endgroup$ May 5, 2015 at 18:57
  • $\begingroup$ These $P$-partition enumerators and their generalizations are the subject of several papers by Malvenuto and Reutenauer, e.g., arXiv:1407.0476v2, arXiv:0905.3508v1, and "Plethysm and conjugation of quasi-symmetric functions". One thing that generalizes very well from the classical case are formulas for the coproduct and the antipode (in the Hopf algebra $\operatorname{QSym}$). I am trying to write an expository paper about this for a year now... $\endgroup$ May 5, 2015 at 22:56
  • $\begingroup$ More details in this (very recent!) survey: arxiv.org/abs/1505.01115. $\endgroup$ May 6, 2015 at 1:09
  • $\begingroup$ For so-called %-avoiding diagrams, see the paper by Reiner and Shimozono at citeseerx.ist.psu.edu/viewdoc/…. $\endgroup$ May 6, 2015 at 2:34

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