|
|
A321854
|
|
Irregular triangle where T(H(u),H(v)) is the number of ways to partition the Young diagram of u into vertical sections whose sizes are the parts of v, where H is Heinz number.
|
|
10
|
|
|
1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 1, 1, 3, 1, 0, 2, 0, 4, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 6, 0, 6, 1, 1, 3, 4, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,11
|
|
COMMENTS
|
A vertical section is a partial Young diagram with at most one square in each row.
|
|
LINKS
|
|
|
EXAMPLE
|
Triangle begins:
1
1
0 1
1 1
0 0 1
0 2 1
0 0 0 0 1
1 3 1
0 2 0 4 1
0 0 0 3 1
0 0 0 0 0 0 1
0 2 2 5 1
0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 4 1
0 0 0 6 0 6 1
1 3 4 6 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 4 10 4 8 1
The 12th row counts the following partitions of the Young diagram of (211) into vertical sections (shown as colorings by positive integers):
T(12,7) = 0:
.
T(12,9) = 2: 1 2 1 2
1 2
2 1
.
T(12,10) = 2: 1 2 1 2
2 1
2 1
.
T(12,12) = 5: 1 2 1 2 1 2 1 2 1 2
3 2 3 1 3
3 3 2 3 1
.
T(12,16) = 1: 1 2
3
4
|
|
MATHEMATICA
|
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
spsu[_, {}]:={{}}; spsu[foo_, set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsu[Select[foo, Complement[#, Complement[set, s]]=={}&], Complement[set, s]]]/@Cases[foo, {i, ___}];
ptnpos[y_]:=Position[Table[1, {#}]&/@y, 1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]], UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]}, Table[Length[Select[spsu[ptnverts[y], ptnpos[y]], Sort[Length/@#]==primeMS[k]&]], {k, Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}]], {n, 18}]
|
|
CROSSREFS
|
Cf. A000085, A000110, A007016, A056239, A122111, A153452, A215366, A296188, A300121, A318396, A321719-A321731, A321737, A321738, A321742-A321765.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|