# Planar Partitions EIS=A000219

vdmcc w.meeussen.vdmcc at vandemoortele.be
Mon May 17 18:11:40 CEST 1999

```Safe Pilings of boxes in a corner. (cfr Planar Partitions EIS=A000219)

Imagine a room, where boxes (cubes) are to be safely piled in the far left
corner.
For stability reasons, stacking height should not increase away from the
corner.
Flapping down the zy- backwall and the xz-left wall, this looks like :
z
1
1
2 2
3 2 2 1 1 y

z 1 1 2 5   4 2 1 1 1 y
2 3   2 2 1
1   1
x   x

where each integer indicates the stacking height of boxes (3D-Ferrers
diagram).
From any projection, both other projections can be easily constructed.
----------------------------
if we write the example above as {{4,2,1,1,1},{2,2,1},{1}}, then
all 24 safe pilings of 5 boxes are:
Flatten[pilings /@ Partitions[5],1]//ColumnForm

{{5}}
{{4, 1}}
{{3, 2}}
{{3, 1, 1}}
{{2, 2, 1}}
{{2, 1, 1, 1}}
{{1, 1, 1, 1, 1}}
{{4}, {1}}
{{3, 1}, {1}}
{{2, 2}, {1}}
{{2, 1, 1}, {1}}
{{1, 1, 1, 1}, {1}}
{{3}, {2}}
{{2, 1}, {2}}
{{2, 1}, {1, 1}}
{{1, 1, 1}, {1, 1}}
{{3}, {1}, {1}}
{{2, 1}, {1}, {1}}
{{1, 1, 1}, {1}, {1}}
{{2}, {2}, {1}}
{{1, 1}, {1, 1}, {1}}
{{2}, {1}, {1}, {1}}
{{1, 1}, {1}, {1}, {1}}
{{1}, {1}, {1}, {1}, {1}}
----------------------------
in how many ways can n boxes be piled?

Table[Plus@@(Length[pilings[#]]&/@Partitions[w]),{w,16}]
{1,3,6,13,24,48,86,160,282,500,859,1479,2485,4167,6879,11297}

Q: in how many ways, taking x-y reflections as equivalent?
Q: in how many ways, taking x-y-z-x rotations as equivalent?
Q: in how many ways, taking both symmetries as equivalent?
Is there a 'smart' way to describe these "safe pilings" using the
standard tools of combinatorics (permutations, combinations, KSubsets,
partitions, Y-Tableaux and the rest)?
-----------------------------
Consider the total number of boxes
in the planes z=1, 2, .. = {9,4,1,1}
in the planes x=1, 2, ..  ={9,5,1} ;
in the planes y=1, 2, ..  ={7,4,2,1,1};
each are partitions of 15.
Do any two of these partitions-of-15 determine the third?
no, because (counterexample)
5 5  and  6 4 both have rowsums {10,5} and columnsums {9,6}
4 1       3 2
in the first case, the z-sums are {4,3,3,3,2} while in the
second case, it is {4,4,3,2,1,1}

------------------------------
Now, in the spirit of counting the number of such pilings of 15 boxes,
let us indicate with a counter (in all three projections) where a 16th box
can be put:
4
.
6 3
8 . 2
9 7 . 5 . 1
4 . 3 2 1   4 3 2 . . 1
6 . 5   6 . . 5
8 7   8 7
9   9

evidently, this pile can safely be extended in 9 ways.
------------------------------
for my archive:
pilings[par_List]:=
Module[{},tel=Length[par];all= List/@Partitions[par[[1]] ];

all=
ten[
Function[parit,
Flatten[{parit,{#}},1]& /@(
Select[Partitions[
par[[i]] ],(
Length[#]<=Length[Last at parit]&&
all,1]
,{i,2,tel}];all
]
----------------------------------------
ID Number: A000219 (Formerly M2566 and N1016)
Sequence:  1,1,3,6,13,24,48,86,160,282,500,859,1479,2485,4167,6879,11297,18334,
29601,47330,75278,118794,186475,290783,451194,696033,1068745,1632658,
2483234,3759612,5668963,8512309,12733429,18974973,28175955
Name:      Planar partitions of n.
References A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for
m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol.
1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman,
ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145,

eq. (1.6).
Knuth, D. E.; A Note on Solid Partitions. Math. Comp. 24,
955-961, 1970.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge,
MA: MIT Artificial
Intelligence Laboratory, Memo AIM-239, Item 18, Feb. 1972.
Bender, E. A. and Knuth, D. E. ``Enumeration of Plane
Partitions.'' J. Combin. Theory A.
13, 40-54, 1972.
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p.
241.
Formula:   G.f.: prod from k = 1 to inf ( 1 - x^k )^(-k).Keywords:
nonn,nice
Offset:    0
Author(s): njas

w.meeussen.vdmcc at vandemoortele.be
tel  +32 (0) 51 33 21 11
fax +32 (0) 51 33 21 75

```