A325190 Number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 2.

0, 0, 2, 0, 0, 2, 4, 2, 2, 4, 8, 10, 12, 10, 14, 20, 28, 36, 44, 46, 56, 66, 86, 108, 136, 160, 190, 214, 252, 298, 364, 434, 524, 620, 728, 834, 966, 1112, 1306, 1522, 1788, 2088, 2448, 2822, 3256, 3720, 4264, 4876, 5610, 6434, 7420

Offset: 0

Author

Gus Wiseman, Apr 11 2019

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A325186.

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. For example, the partition (6,5,5,3) has 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).

Examples

			The a(2) = 1 through a(11) = 10 partitions:
  (2)   (32)   (33)    (52)     (62)      (72)       (82)        (92)
  (11)  (221)  (42)    (22111)  (221111)  (432)      (433)       (443)
               (222)                      (3321)     (442)       (533)
               (2211)                     (2211111)  (532)       (542)
                                                     (3322)      (632)
                                                     (3331)      (3332)
                                                     (33211)     (33221)
                                                     (22111111)  (33311)
                                                                 (332111)
                                                                 (221111111)
		

Links

• Eric Weisstein's World of Mathematics, Graph Distance.

Crossrefs

Cf. A188674, A325165, A325169, A325183, A325184, A325186, A325187, A325190, A325199.

Programs

Mathematica

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[Length[Select[IntegerPartitions[n],Apply[Plus,If[#=={},{},FixedPointList[corpos,ptnmat[#]][[-3]]],{0,1}]==2&]],{n,30}]