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A325187
Number of integer partitions of n such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.
11
1, 0, 1, 3, 3, 5, 9, 14, 20, 26, 38, 53, 75, 101, 132, 175, 229, 301, 394, 509, 650, 826, 1043, 1315, 1656, 2074, 2590, 3218, 3975, 4896, 6008, 7361, 8989, 10960, 13323, 16159, 19531, 23553, 28323, 34002, 40723, 48694, 58115, 69249, 82350, 97766, 115832
OFFSET
1,4
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 sequence gives the number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 1.
The Heinz numbers of these partitions are given by A325185.
LINKS
Eric Weisstein's World of Mathematics, Graph Distance.
EXAMPLE
The a(1) = 1 through a(8) = 14 partitions:
(1) (21) (22) (41) (51) (61) (71)
(31) (311) (321) (322) (332)
(211) (2111) (411) (331) (422)
(3111) (421) (431)
(21111) (511) (521)
(3211) (611)
(4111) (3221)
(31111) (3311)
(211111) (4211)
(5111)
(32111)
(41111)
(311111)
(2111111)
MATHEMATICA
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
Table[Length[Select[IntegerPartitions[n], otb[#]>otb[Rest[#]]&&otb[#]>otb[DeleteCases[#-1, 0]]&]], {n, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 11 2019
STATUS
approved