A Generalization of Macmahon’s Formula
نویسندگان
چکیده
We generalize the generating formula for plane partitions known as MacMahon’s formula as well as its analog for strict plane partitions. We give a 2-parameter generalization of these formulas related to Macdonald’s symmetric functions. The formula is especially simple in the Hall-Littlewood case. We also give a bijective proof of the analog of MacMahon’s formula for strict plane partitions.
منابع مشابه
Plane Partitions I: a Generalization of Macmahon’s Formula
The number of plane partitions contained in a given box was shown by MacMahon [8] to be given by a simple product formula. By a simple bijection, this formula also enumerates lozenge tilings of hexagons of side-lengths a, b, c, a, b, c (in cyclic order) and angles of 120 degrees. We present a generalization in the case b = c by giving simple product formulas enumerating lozenge tilings of regio...
متن کاملA dual of MacMahon’s theorem on plane partitions
A classical theorem of MacMahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula. In this paper, we present a counterpart of this formula, corresponding to the exterior of a concave hexagon obtained by turning 120° after drawing each side (MacMahon’s hexagon is obtained by turning 60° af...
متن کاملShifted Schur Process and Asymptotics of Large Random Strict Plane Partitions
In this paper we define the shifted Schur process as a measure on sequences of strict partitions. This process is a generalization of the shifted Schur measure introduced in [TW] and [Mat] and is a shifted version of the Schur process introduced in [OR1]. We prove that the shifted Schur process defines a Pfaffian point process. We further apply this fact to compute the bulk scaling limit of the...
متن کاملElementary Proof of MacMahon’s Conjecture
Major Percy A. MacMahon’s first paper on plane partitions [4] included a conjectured generating function for symmetric plane partitions. This conjecture was proven almost simultaneously by George Andrews and Ian Macdonald, Andrews using the machinery of basic hypergeometric series [1] and Macdonald employing his knowledge of symmetric functions [3]. The purpose of this paper is to simplify Macd...
متن کاملMACMAHON’S PARTITION ANALYSIS IX: k-GON PARTITIONS
MacMahon devoted a significant portion of Volume II of his famous book “Combinatory Analysis” to the introduction of Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. In a series of papers we have shown that MacMahon’s method turns into an extremely powerful tool when implemented in comput...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007