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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..47.

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}]

CROSSREFS

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

Sequence in context: A325849 A104220 A321986 * A209083 A137202 A146926

Adjacent sequences:  A325184 A325185 A325186 * A325188 A325189 A325190

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 11 2019

STATUS

approved

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Last modified January 20 23:29 EST 2020. Contains 331104 sequences. (Running on oeis4.)