A325184 Last part of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.

0, 1, 2, 2, 3, 1, 4, 3, 1, 1, 5, 1, 6, 1, 2, 4, 7, 2, 8, 1, 2, 1, 9, 1, 2, 1, 2, 1, 10, 1, 11, 5, 2, 1, 3, 2, 12, 1, 2, 1, 13, 1, 14, 1, 1, 1, 15, 1, 3, 1, 2, 1, 16, 3, 3, 1, 2, 1, 17, 1, 18, 1, 1, 6, 3, 1, 19, 1, 2, 1, 20, 2, 21, 1, 1, 1, 4, 1, 22, 1, 3, 1

Offset

1,3

Comments

The k-th part of the origin-to-boundary partition of a Young diagram is the number of squares graph-distance k from the lower-right boundary.

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

Links

Table of n, a(n) for n=1..82.

Eric Weisstein's World of Mathematics, Graph Distance.

Example

The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with origin-to-boundary graph-distances
  4 4 4 3 2 1
  3 3 3 2 1
  2 2 2 1 1
  1 1 1
giving the origin-to-boundary partition (7,5,4,3) with last part 3, so a(7865) = 3.

Mathematica

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
ptnmat[ptn_]:=PadRight[(ConstantArray[1, #]&)/@Sort[ptn, Greater], {Length[ptn], Max@@ptn}+1];
corpos[mat_]:=ReplacePart[mat, Select[Position[mat, 1], Times@@Extract[mat, {#+{1, 0}, #+{0, 1}}]==0&]->0];
Table[Apply[Plus, If[n==1, {}, FixedPointList[corpos, ptnmat[primeptn[n]]][[-3]]], {0, 1}], {n, 100}]

Crossrefs

Positions of 1's are A325185. Positions of 2's are A325186.

Cf. A056239, A065770, A093641, A112798, A174090, A325166, A325169, A325183, A325187.

Sequence in context: A207507 A278537 A331803 * A353931 A354871 A303777

Adjacent sequences: A325181 A325182 A325183 * A325185 A325186 A325187

Keyword

nonn

Author

Gus Wiseman, Apr 08 2019

Status

approved