The Equivalence between Enumerating Cyclically Symmetric, Self-Complementary and Totally Symmetric, Self-Complementary Plane Partitions
نویسنده
چکیده
We prove that the number of cyclically symmetric, self-complementary plane partitions contained in a cube of side 2n equals the square of the number of totally symmetric, self-complementary plane partitions contained in the same cube, without explicitly evaluating either of these numbers. This appears to be the first direct proof of this fact. The problem of finding such a proof was suggested by Stanley [9].
منابع مشابه
(−1)–enumeration of Plane Partitions with Complementation Symmetry
We compute the weighted enumeration of plane partitions contained in a given box with complementation symmetry where adding one half of an orbit of cubes and removing the other half of the orbit changes the weight by −1 as proposed by Kuperberg in [7, pp.25/26]. We use nonintersecting lattice path families to accomplish this for transpose–complementary, cyclically symmetric transpose–complement...
متن کاملA Connection between Alternating Sign Matrices and Totally Symmetric Self-Complementary Plane Partitions
We give a lattice path interpretation for totally symmetric self-complementary plane partitions. This is a first step in solving the long standing problem of enumerating such plane partitions. Another outstanding problem in enumerative combinatorics is the search for a bijection between alternating sign matrices and totally symmetric self-complementary plane partitions. From the lattice path in...
متن کاملOn Powers of 2 Dividing the Values of Certain Plane Partition Functions
We consider two families of plane partitions: totally symmetric self-complementary plane partitions (TSSCPPs) and cyclically symmetric transpose complement plane partitions (CSTCPPs). If T (n) and C(n) are the numbers of such plane partitions in a 2n× 2n× 2n box, then ord2(T (n)) = ord2(C(n)) for all n ≥ 1. We also discuss various consequences, along with other results on ord2(T (n)). 2000 Math...
متن کاملOn refined enumerations of totally symmetric self-complementary plane partitions I
Abstract In this paper we give Pfaffian expressions and constant term identities for three conjectures (i.e. Conjecture 2, Conjecture 3 and Conjecture 7) by Mills, Robbins and Rumsey in the paper “Self-complementary totally symmetric plane partitions” J. Combin. Theory Ser. A 42, 277–292) concerning the refined enumeration problems of totally symmetric self-complementary plane partitions. We al...
متن کاملRefined Enumerations of Totally Symmetric Self-Complementary Plane Partitions and Lattice Path Combinatorics
This article is a short explanation of some of the results obtained in my papers “On refined enumerations of totally symmetric self-complementary plane partitions I, II”. We give Pfaffian expressions for some of the conjectures in the paper “Self-complementary totally symmetric plane partitions” (J. Combin. Theory Ser. A 42, 277–292) by Mills, Robbins and Rumsey, using the lattice path method.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 86 شماره
صفحات -
تاریخ انتشار 1999