A325200 -id:A325200 - cp's OEIS frontend

A325199 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.

0, 0, 0, 2, 0, 2, 6, 3, 2, 9, 15, 12, 6, 12, 27, 38, 34, 22, 20, 43, 74, 94, 90, 67, 48, 69, 130, 194, 232, 230, 187, 132, 129, 218, 364, 497, 576, 578, 498, 367, 290, 378, 642, 977, 1264, 1435, 1448, 1290, 1000, 735, 728

Offset: 0

Gus Wiseman, Apr 11 2019

The Heinz numbers of these partitions are given by A325197.

			The a(3) = 2 through a(10) = 15 partitions (empty columns not shown):
  (3)    (41)    (33)    (43)    (521)    (333)    (433)
  (111)  (2111)  (42)    (2221)  (32111)  (441)    (442)
                 (222)   (4111)           (522)    (532)
                 (411)                    (531)    (541)
                 (2211)                   (3222)   (3322)
                 (3111)                   (5211)   (3331)
                                          (32211)  (4222)
                                          (33111)  (4411)
                                          (42111)  (5221)
                                                   (5311)
                                                   (32221)
                                                   (33211)
                                                   (42211)
                                                   (43111)
                                                   (52111)

Column k=2 of A325200.

Cf. A046660, A065770, A071724, A243055, A325166, A325169, A325178, A325188, A325189, A325191, A325195, A325197, A325198.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otbmax[#]-otb[#]==2&]],{n,0,30}]

A368986 a(n) = sum of the origin-to-boundary graph-distances of all partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 8, 12, 21, 32, 50, 73, 107, 152, 219, 302, 419, 567, 771, 1027, 1374, 1806, 2375, 3083, 3999, 5136, 6597, 8398, 10676, 13477, 16981, 21260, 26584, 33057, 41049, 50738, 62605, 76930, 94374, 115330, 140704, 171106, 207732, 251460, 303919, 366335, 440880, 529298

Offset: 0

Andrew Howroyd, Jan 12 2024

nonn

The origin-to-boundary graph-distance (see A325188) is the side length of the maximum triangular partition contained inside the Ferrer's diagram of the partition. a(n) is the sum of the side lengths over all partitions of n.

Cf. A325188, A325189, A325200.

a(n)={my(s=0); forpart(p=n, my(w=#p); for(i=1, #p, w=min(w, #p-i+p[i])); s += w); s}

a(n) = Sum_{k=1..n} k*A325188(n,k).

cp's OEIS Frontend

A325199 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.

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