scholarly journals The Enumeration of Totally Symmetric Plane Partitions

1995 ◽  
Vol 111 (2) ◽  
pp. 227-243 ◽  
Author(s):  
J.R. Stembridge
Keyword(s):  
Plane Partitions ◽  
Symmetric Plane

A NEW CONGRUENCE MODULO 25 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

2014 ◽  
Vol 91 (1) ◽  
pp. 41-46 ◽  
Author(s):  
ERNEST X. W. XIA
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Plane Partitions ◽  
Arithmetic Properties ◽  
Symmetric Plane ◽  
Congruence Modulo

AbstractFor any positive integer $n$, let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of $n$. Recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.89 (2014), 473–478] and Yao [‘New infinite families of congruences modulo 4 and 8 for 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.90 (2014), 37–46] proved a number of congruences satisfied by $f(n)$. In particular, Hirschhorn and Sellers proved that $f(10n+5)\equiv 0\ (\text{mod}\ 5)$. In this paper, we establish the generating function of $f(30n+25)$ and prove that $f(250n+125)\equiv 0\ (\text{mod\ 25}).$

scholarly journals Cyclic Sieving for Plane Partitions and Symmetry

2020 ◽  
Author(s):  
Sam Hopkins ◽  
Keyword(s):  
Fixed Points ◽  
Generating Function ◽  
Cyclic Group ◽  
Product Formula ◽  
Roots Of Unity ◽  
Plane Partitions ◽  
Cyclic Sieving Phenomenon ◽  
Combinatorial Set ◽  
Symmetric Plane ◽  
Cyclic Sieving

The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.

scholarly journals Volume Laws for Boxed Plane Partitions and Area Laws for Ferrers Diagrams

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Uwe Schwerdtfeger
Keyword(s):  
Uniform Distribution ◽  
Random Variables ◽  
Symmetry Class ◽  
Limit Laws ◽  
Plane Partitions ◽  
Symmetric Plane ◽  
International Audience ◽  
Ferrers Diagrams ◽  
Area Limit

International audience We asymptotically analyse the volume random variables of general, symmetric and cyclically symmetric plane partitions fitting inside a box. We consider the respective symmetry class equipped with the uniform distribution. We also prove area limit laws for two ensembles of Ferrers diagrams. Most limit laws are Gaussian.

scholarly journals ARITHMETIC PROPERTIES OF 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

2013 ◽  
Vol 89 (3) ◽  
pp. 473-478 ◽  
Author(s):  
MICHAEL D. HIRSCHHORN ◽  
JAMES A. SELLERS
Keyword(s):  
Generating Function ◽  
Main Diagonal ◽  
Plane Partitions ◽  
Combinatorial Objects ◽  
Arithmetic Properties ◽  
Symmetric Plane ◽  
Image Position

AbstractBlecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’,Util. Math. 88(2012), 223–235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight$n$. He also proved that the generating function for$f(n), $the number of 1-shell totally symmetric plane partitions of weight$n$, is given by$$\begin{eqnarray*}\displaystyle \sum _{n\geq 0}f(n){q}^{n} = 1+ \sum _{n\geq 1}{q}^{3n- 2} \prod _{i= 0}^{n- 2} (1+ {q}^{6i+ 3} ).\end{eqnarray*}$$In this brief note, we prove a number of arithmetic properties satisfied by$f(n)$using elementary generating function manipulations and well-known results of Ramanujan and Watson.

scholarly journals The 2-adic Valuation of Plane Partitions and Totally Symmetric Plane Partitions

10.37236/2064 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
William J. Keith
Keyword(s):  
Plane Partitions ◽  
Symmetric Plane ◽  
Adic Valuation

This paper confirms a conjecture of Amdeberhan and Moll that the power of 2 dividing the number of plane partitions in an $n$-cube is greater than the power of 2 dividing the number of totally symmetric plane partitions in the same cube when $n$ is even, and less when $n$ is odd.

scholarly journals The poset perspective on alternating sign matrices

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jessica Striker
Keyword(s):  
Generating Function ◽  
Young Tableaux ◽  
Alternating Sign Matrices ◽  
Plane Partitions ◽  
Combinatorial Objects ◽  
Symmetric Plane ◽  
International Audience ◽  
Order Ideals ◽  
Nous Utilisons

International audience Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

scholarly journals Arithmetic Properties of Plane Partitions

10.37236/1997 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Tewodros Amdeberhan ◽  
Victor H. Moll
Keyword(s):  
Simple Manner ◽  
Alternating Sign Matrices ◽  
Plane Partitions ◽  
Arithmetic Properties ◽  
Symmetric Plane

The $2$-adic valuations of sequences counting the number of alternating sign matrices of size $n$ and the number of totally symmetric plane partitions are shown to be related in a simple manner.

scholarly journals Self-complementary totally symmetric plane partitions

1986 ◽  
Vol 42 (2) ◽  
pp. 277-292 ◽  
Author(s):  
W.H Mills ◽  
David P Robbins ◽  
Howard Rumsey
Keyword(s):  
Plane Partitions ◽  
Symmetric Plane
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