A321738 Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections.

1, 1, 1, 2, 1, 3, 1, 5, 7, 4, 1, 10, 1, 5, 13, 15, 1, 27, 1, 17, 21, 6, 1, 37, 34, 7, 87, 26, 1, 60, 1, 52, 31, 8, 73, 114, 1, 9, 43, 77, 1, 115, 1, 37, 235, 10, 1, 151, 209, 175, 57, 50, 1, 409, 136, 141, 73, 11, 1, 295, 1, 12, 543, 203, 229, 198, 1, 65, 91

Offset

1,4

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:

1 2 3

1 2

2 3

Links

Table of n, a(n) for n=1..69.

Example

The a(12) = 10 partitions of the Young diagram of (211) into vertical sections:
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3     3     2     3     2     1     1     3     2     1
  4     3     3     2     2     3     2     1     1     1

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]]}, Length[spsu[ptnverts[y], ptnpos[y]]]], {n, 30}]

Crossrefs

Cf. A000110, A000700, A000701, A006052, A056239, A122111, A320328, A321719-A321731, A321737, A321854.

Sequence in context: A336617 A341865 A347982 * A022458 A084419 A119606

Adjacent sequences: A321735 A321736 A321737 * A321739 A321740 A321741

Keyword

nonn,more

Author

Gus Wiseman, Nov 19 2018

Status

approved