A325166 -id:A325166 - cp's OEIS frontend

A325167 Heinz number of the internal portion of the integer partition with Heinz number n.

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 3, 1, 1, 4, 1, 2, 3, 2, 1, 2, 3, 2, 4, 2, 1, 6, 1, 1, 3, 2, 5, 4, 1, 2, 3, 2, 1, 6, 1, 2, 6, 2, 1, 2, 5, 6, 3, 2, 1, 8, 5, 2, 3, 2, 1, 6, 1, 2, 6, 1, 5, 6, 1, 2, 3, 10, 1, 4, 1, 2, 9, 2, 7, 6, 1, 2, 8, 2, 1, 6, 5, 2, 3

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The internal portion of an integer partition consists of all squares in the Young diagram that have a square both directly below and directly to the right.

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

			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 internal portion
  o o o o o
  o o o o
  o o o
which is the partition (5,4,3), with Heinz number 385, so a(7865) = 385.

Cf. A052126, A056239, A064989, A065770, A093641, A112798, A257990, A297113, A325133, A325166, A325169.

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

Original entry on oeis.org

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

Gus Wiseman, Apr 08 2019

nonn

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).

			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.

Eric Weisstein's World of Mathematics, Graph Distance.

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

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

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}]

A325199 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 6, 3, 2, 9, 15, 12, 6, 12, 27, 38, 34, 22, 20, 43, 74, 94, 90, 67, 48, 69, 130, 194, 232, 230, 187, 132, 129, 218, 364, 497, 576, 578, 498, 367, 290, 378, 642, 977, 1264, 1435, 1448, 1290, 1000, 735, 728

Offset: 0

Gus Wiseman, Apr 11 2019

The Heinz numbers of these partitions are given by A325197.

			The a(3) = 2 through a(10) = 15 partitions (empty columns not shown):
  (3)    (41)    (33)    (43)    (521)    (333)    (433)
  (111)  (2111)  (42)    (2221)  (32111)  (441)    (442)
                 (222)   (4111)           (522)    (532)
                 (411)                    (531)    (541)
                 (2211)                   (3222)   (3322)
                 (3111)                   (5211)   (3331)
                                          (32211)  (4222)
                                          (33111)  (4411)
                                          (42111)  (5221)
                                                   (5311)
                                                   (32221)
                                                   (33211)
                                                   (42211)
                                                   (43111)
                                                   (52111)

Column k=2 of A325200.

Cf. A046660, A065770, A071724, A243055, A325166, A325169, A325178, A325188, A325189, A325191, A325195, A325197, A325198.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otbmax[#]-otb[#]==2&]],{n,0,30}]

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A325167 Heinz number of the internal portion of the integer partition with Heinz number n.

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

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A325199 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.

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