Four factorization formulas for plane partitions

Ciucu, Mihai

doi:10.1090/proc/12800

Four factorization formulas for plane partitions
HTML articles powered by AMS MathViewer

by Mihai Ciucu PDF

Proc. Amer. Math. Soc. 144 (2016), 1841-1856 Request permission

Abstract:

All ten symmetry classes of plane partitions that fit in a given box are known to be enumerated by simple product formulas, but there is still no unified proof for all of them. Progress towards this goal can be made by establishing identities connecting the various symmetry classes. We present in this paper four such identities, involving all ten symmetry classes. We discuss their proofs and generalizations. The main result of this paper is to give a generalization of one of them, in the style of the identity presented in “A factorization theorem for rhombus tilings,” M. Ciucu and C. Krattenthaler, arXiv:1403.3323.

References

George E. Andrews, Plane partitions. V. The TSSCPP conjecture, J. Combin. Theory Ser. A 66 (1994), no. 1, 28–39. MR 1273289, DOI 10.1016/0097-3165(94)90048-5
Mihai Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67–97. MR 1426739, DOI 10.1006/jcta.1996.2725
Mihai Ciucu, The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions, J. Combin. Theory Ser. A 86 (1999), no. 2, 382–389. MR 1685538, DOI 10.1006/jcta.1998.2922
M. Ciucu, Plane partitions I: A generalization of MacMahon’s formula, Mem. Amer. Math. Soc. 178 (2005), no. 839, 107–144.
Mihai Ciucu, Lozenge tilings with gaps in a $90^\circ$ wedge domain with mixed boundary conditions, Comm. Math. Phys. 334 (2015), no. 1, 507–532. MR 3304284, DOI 10.1007/s00220-014-2138-2
M. Ciucu and C. Krattenthaler, A factorization theorem for rhombus tilings, preprint (2014), arXiv:1403.3323.
P. A. MacMahon, Memoir on the theory of the partition of numbers—Part V. Partitions in two-dimensional space, Phil. Trans. R. S., 1911, A.
Guy David and Carlos Tomei, The problem of the calissons, Amer. Math. Monthly 96 (1989), no. 5, 429–431. MR 994034, DOI 10.2307/2325150
Greg Kuperberg, Symmetries of plane partitions and the permanent-determinant method, J. Combin. Theory Ser. A 68 (1994), no. 1, 115–151. MR 1295786, DOI 10.1016/0097-3165(94)90094-9
Richard P. Stanley, Symmetries of plane partitions, J. Combin. Theory Ser. A 43 (1986), no. 1, 103–113. MR 859302, DOI 10.1016/0097-3165(86)90028-2
John R. Stembridge, The enumeration of totally symmetric plane partitions, Adv. Math. 111 (1995), no. 2, 227–243. MR 1318529, DOI 10.1006/aima.1995.1023