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A325190
Number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 2.
3
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
0,3
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).
LINKS
Eric Weisstein's World of Mathematics, Graph Distance.
EXAMPLE
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)
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 11 2019
STATUS
approved