Digital Library of Mathematical Functions
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26 Combinatorial AnalysisProperties

§26.12 Plane Partitions

Contents

§26.12(i) Definitions

A plane partition, π, of a positive integer n, is a partition of n in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. Different configurations are counted as different plane partitions. As an example, there are six plane partitions of 3:

26.12.1 3,
21,
21,
111,
111,
111.

An equivalent definition is that a plane partition is a finite subset of ×× with the property that if (r,s,t)π and (1,1,1)(h,j,k)(r,s,t), then (h,j,k) must be an element of π. Here (h,j,k)(r,s,t) means hr, js, and kt. It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point (h,j,k)π. For example, Figure 26.12.1 depicts the pile of blocks that represents the plane partition of 75 given by (26.12.2).

See accompanying text
Figure 26.12.1: A plane partition of 75. Magnify
26.12.2 65543364331643114221311111

The number of plane partitions of n is denoted by pp(n), with pp(0)=1. See Table 26.12.1.

Table 26.12.1: Plane partitions.
n pp(n) n pp(n) n pp(n)
0 1 17 18334 34 281 75955
1 1 18 29601 35 416 91046
2 3 19 47330 36 614 84961
3 6 20 75278 37 903 79784
4 13 21 1 18794 38 1324 41995
5 24 22 1 86475 39 1934 87501
6 48 23 2 90783 40 2818 46923
7 86 24 4 51194 41 4093 83981
8 160 25 6 96033 42 5930 01267
9 282 26 10 68745 43 8566 67495
10 500 27 16 32658 44 12343 63833
11 859 28 24 83234 45 17740 79109
12 1479 29 37 59612 46 25435 35902
13 2485 30 56 68963 47 36379 93036
14 4167 31 85 12309 48 51913 04973
15 6879 32 127 33429 49 73910 26522
16 11297 33 189 74973 50 1 04996 40707

We define the r×s×t box B(r,s,t) as

26.12.3 B(r,s,t)={(h,j,k)| 1hr,1js,1kt}.

Then the number of plane partitions in B(r,s,t) is

26.12.4 (h,j,k)B(r,s,t)h+j+k-1h+j+k-2=h=1rj=1sh+j+t-1h+j-1.

A plane partition is symmetric if (h,j,k)π implies that (j,h,k)π. The number of symmetric plane partitions in B(r,r,t) is

26.12.5 h=1r2h+t-12h-11h<jrh+j+t-1h+j-1.

A plane partition is cyclically symmetric if (h,j,k)π implies (j,k,h)π. The plane partition in Figure 26.12.1 is an example of a cyclically symmetric plane partition. The number of cyclically symmetric plane partitions in B(r,r,r) is

26.12.6 h=1r3h-13h-21h<jrh+2j-1h+j-1,

or equivalently,

26.12.7 h=1r(3h-13h-2j=hrr+h+j-12h+j-1).

A plane partition is totally symmetric if it is both symmetric and cyclically symmetric. The number of totally symmetric plane partitions in B(r,r,r) is

26.12.8 1hjrh+j+r-1h+2j-2.

The complement of πB(r,s,t) is πc={(h,j,k)| (r-h+1,s-j+1,t-k+1)π}. A plane partition is self-complementary if it is equal to its complement. The number of self-complementary plane partitions in B(2r,2s,2t) is

26.12.9 (h=1rj=1sh+j+t-1h+j-1)2;

in B(2r+1,2s,2t) it is

26.12.10 (h=1rj=1sh+j+t-1h+j-1)(h=1r+1j=1sh+j+t-1h+j-1);

in B(2r+1,2s+1,2t) it is

26.12.11 (h=1r+1j=1sh+j+t-1h+j-1)(h=1rj=1s+1h+j+t-1h+j-1).

A plane partition is transpose complement if it is equal to the reflection through the (x,y)-plane of its complement. The number of transpose complement plane partitions in B(r,r,2t) is

26.12.12 (t+r-1r-1)1hjr-2h+j+2t+1h+j+1.

The number of symmetric self-complementary plane partitions in B(2r,2r,2t) is

26.12.13 h=1rj=1rh+j+t-1h+j-1;

in B(2r+1,2r+1,2t) it is

26.12.14 h=1rj=1r+1h+j+t-1h+j-1.

The number of cyclically symmetric transpose complement plane partitions in B(2r,2r,2r) is

26.12.15 h=0r-1(3h+1)(6h)!(2h)!(4h+1)!(4h)!.

The number of cyclically symmetric self-complementary plane partitions in B(2r,2r,2r) is

26.12.16 (h=0r-1(3h+1)!(r+h)!)2.

The number of totally symmetric self-complementary plane partitions in B(2r,2r,2r) is

26.12.17 h=0r-1(3h+1)!(r+h)!.

A strict shifted plane partition is an arrangement of the parts in a partition so that each row is indented one space from the previous row and there is weak decrease across rows and strict decrease down columns. An example is given by:

26.12.18 66643332

A descending plane partition is a strict shifted plane partition in which the number of parts in each row is strictly less than the largest part in that row and is greater than or equal to the largest part in the next row. The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition. The number of descending plane partitions in B(r,r,r) is

26.12.19 h=0r-1(3h+1)!(r+h)!.

§26.12(ii) Generating Functions

The notation πB(r,s,t) denotes the sum over all plane partitions contained in B(r,s,t), and |π| denotes the number of elements in π.

26.12.20 π××q|π|=k=11(1-qk)k,
26.12.21 πB(r,s,t)q|π|=(h,j,k)B(r,s,t)1-qh+j+k-11-qh+j+k-2=h=1rj=1s1-qh+j+t-11-qh+j-1,
26.12.22 πB(r,r,t)π symmetricq|π|=h=1r1-q2h+t-11-q2h-11h<jr1-q2(h+j+t-1)1-q2(h+j-1).
26.12.23 πB(r,r,r)π cyclically symmetricq|π|=h=1r1-q3h-11-q3h-21h<jr1-q3(h+2j-1)1-q3(h+j-1)=h=1r(1-q3h-11-q3h-2j=hr1-q3(r+h+j-1)1-q3(2h+j-1)).
26.12.24 πB(r,r,r)π descending plane partitionq|π|=1h<jr1-qr+h+j-11-q2h+j-1.

§26.12(iii) Recurrence Relation

26.12.25 pp(n)=1nj=1npp(n-j)σ2(j),

where σ2(j) is the sum of the squares of the divisors of j.

§26.12(iv) Limiting Form

As n

26.12.26 pp(n)(ζ(3))7/36211/36(3π)1/2n25/36exp(3(ζ(3))1/3(12n)2/3+ζ(-1)),

where ζ is the Riemann ζ-function (§25.2(i)).

Addendum: To ten decimal places,

26.12.27 ζ(3) =1.20205 69032,
ζ(-1) =-0.16542 11437.