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.

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

Author

Gus Wiseman, Apr 11 2019

Keywords

• nonn

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.

Examples

			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)
		

Links

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

Crossrefs

Cf. A000245, A188674, A325165, A325169, A325183, A325184, A325185, A325187, A325190, A325191.

Programs

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}]