Plane partitions

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#Plane partitions

A plane partition is an array \(\pi_{ij}\) of non-negative integers with weakly decreasing rows and columns. We let \(|\pi|\) denote the total sum of the entries.

See Stanley’s survey for a bit more background.

Theorem (MacMahon, [Mac]).

We have \[\sum_{\pi} q^{|\pi|} = \prod_{i=1}^a \prod_{j=1}^b \prod_{k=1}^m \frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}\] where the sum is over all plane partitions with \(\leq a\) rows, \(\leq b\) columns and largest entry \(\leq m.\)

Example

For \(a=1,\) \(b=2,\) and \(m=2,\) the plane partitions are weakly decreasing rows \((u,v)\) with \(2\geq u\geq v\geq 0.\) They are \[(0,0),\quad (1,0),\quad (1,1),\quad (2,0),\quad (2,1),\quad (2,2),\] so the generating polynomial is \[1+q+2q^2+q^3+q^4.\] The product formula gives the same value: \[\frac{1-q^2}{1-q} \frac{1-q^3}{1-q^2} \frac{1-q^3}{1-q^2} \frac{1-q^4}{1-q^3} = \frac{(1-q^3)(1-q^4)}{(1-q)(1-q^2)} = 1+q+2q^2+q^3+q^4.\]

Theorem (Kreweras, [Kre65]).

The number of plane partitions of shape \(\lambda/\mu\) with maximal entry at most \(m,\) is given by \[\det\left( \binom{\lambda_i-\mu_j + m}{i-j+m} \right)_{i,j=1,\dotsc,\ell}\] where \(\ell = \length(\lambda).\)

Note that this allows us to efficiently compute the Ehrhart polynomial of order polytopes associated with Ferrers diagrams.

I. Fischer and F. Schreier-Aigner introduce a family of Schur-positive symmetric functions defined as sums over totally symmetric plane partitions [FS24]. For one specialization this family agrees with a multivariate generating function for extended alternating sign matrices. S. Hopkins and T. Lai prove a product formula for plane partitions of shifted double staircase shape [HL21]. I. Pak and F. Petrov study hidden symmetries of weighted lozenge tilings [PP20].

S. Hopkins: Is there a similar formula for shifted shapes?

Bibliography

[FS24]Ilse Fischer and Florian Schreier-Aigner. Alternating sign matrices and totally symmetric plane partitions. Algebraic Combinatorics, 7(5):1319–1345, 2024.

.bib

@article{FischerSchreierAigner2024ASM,
  author = {Ilse Fischer and Florian Schreier-Aigner},
  title = {Alternating sign matrices and totally symmetric plane partitions},
  year = {2024},
  journal = {Algebraic Combinatorics},
  volume = {7},
  number = {5},
  pages = {1319--1345},
  doi = {10.5802/alco.374},
  url = {https://doi.org/10.5802/alco.374},
  eprint = {2201.13142}
}

[HL21]Sam Hopkins and Tri Lai. Plane partitions of shifted double staircase shape. Journal of Combinatorial Theory, Series A, 183, 2021.

.bib

@article{HopkinsLai2021,
  author = {Sam Hopkins and Tri Lai},
  title = {Plane partitions of shifted double staircase shape},
  year = {2021},
  journal = {Journal of Combinatorial Theory, Series A},
  volume = {183},
  doi = {10.1016/j.jcta.2021.105486},
  eprint = {2007.05381}
}

[Kre65]Germain Kreweras. Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers. Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche, 6:9–107, 1965.

.bib

@article{Kreweras1965,
author = {Germain Kreweras},
title = {Sur une classe de probl{\`{e}}mes de d{\'{e}}nombrement li{\'{e}}s au treillis des partitions des entiers},
journal = {Cahiers du Bureau universitaire de recherche op{\'{e}}rationnelle S{\'{e}}rie Recherche},
publisher = {Institut Henri Poincar{\'{e}} - Institut de Statistique de l'Universit{\'{e}} de Paris},
volume = {6},
year = {1965},
pages = {9-107},
language = {fr},
url = {http://www.numdam.org/item/BURO_1965__6__9_0}
}

[Mac]Percy A. MacMahon. Combinatorial analysis, two volumes (bound as one). Chelsea Publishing Co., New York, 1960, .

.bib

@book{MacMahon1960,
	title = {Combinatorial Analysis, Two volumes (bound as one)},
	author = {Percy A. MacMahon},
	publisher = {Chelsea Publishing Co., New York, 1960},
	year = {1915--1916}
}

[PP20]Igor Pak and Fedor Petrov. Hidden symmetries of weighted lozenge tilings. The Electronic Journal of Combinatorics, 27(3), 2020.

.bib

@article{PakPetrov2020,
  author = {Igor Pak and Fedor Petrov},
  title = {Hidden symmetries of weighted lozenge tilings},
  year = {2020},
  journal = {The Electronic Journal of Combinatorics},
  volume = {27},
  number = {3},
  doi = {10.37236/9498},
  eprint = {2003.14236}
}