editing

approved

The 12-th 12th row counts the following partitions of the Young diagram of (211) into vertical sections (shown as colorings by positive integers):

approved

editing

proposed

approved

editing

proposed

Cf. A000085, A000110, A000700, A007016, A056239, A122111, A153452, A215366, A296188, A300121, A318396, A321719-A321731, A321737, A321738, A321742-A321765.

Cf. A000085, A000110, A000700, A007016, A056239, A122111, A153452, A215366, A296188, A300121, A318396, A321719-A321731, A321737, A321738, A321742-A321765.

A vertical section is a partial Young diagram with at most one square in each row.

allocated for Gus WisemanIrregular 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.

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

1,11

Row n has length A000041(A056239(n)).

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 12-th 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

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}]

Cf. A000085, A000110, A000700, A007016, A056239, A122111, A153452, A215366, A296188, A300121, A318396, A321719-A321731, A321737, A321738.

allocated

nonn,tabf

Gus Wiseman, Nov 19 2018

approved

editing

allocated for Gus Wiseman

allocated

approved