cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A325181 Number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 1.

Original entry on oeis.org

0, 0, 2, 1, 0, 2, 3, 2, 1, 0, 2, 3, 4, 3, 2, 1, 0, 2, 3, 4, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 2, 3, 4, 5, 6
Offset: 0

Views

Author

Gus Wiseman, Apr 08 2019

Keywords

Comments

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

Examples

			The a(2) = 2 through a(15) = 1 partitions:
(2)  (21) (32)  (33)  (322) (332) (433)  (443)  (444)  (4333) (4433) (4443)
(11)      (221) (222) (331)       (3331) (3332) (3333) (4432) (4442)
                (321)                    (4331) (4332) (4441)
                                                (4431)

Links

Crossrefs

Cf. A006918, A084835, A096771, A257990, A263297, A325178, A325179, A325182, A325191, A325192, A325198.

Programs

  • Mathematica
    durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==1&]],{n,0,30}]

Extensions

More terms from Giovanni Resta, Apr 15 2019

A325182 Number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 2.

Original entry on oeis.org

0, 0, 0, 2, 2, 1, 2, 4, 7, 6, 5, 4, 5, 9, 12, 15, 14, 12, 10, 9, 11, 15, 21, 24, 28, 26, 24, 20, 18, 17, 19, 25, 31, 38, 42, 46, 44, 41, 36, 32, 29, 28, 31, 37, 46, 53, 62, 66, 71, 68, 65, 58, 53, 47, 44, 43, 46, 54, 63, 74, 83, 93, 98, 103, 100, 96, 88, 81
Offset: 0

Views

Author

Gus Wiseman, Apr 08 2019

Keywords

Comments

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

Examples

			The a(3) = 2 through a(14) = 12 partitions:
  3    31   311  42    43    44    432   442   533    543    544    554
  111  211       2211  421   422   441   3322  4322   4422   553    5333
                       2221  431   3222  4222  4421   5331   5332   5432
                       3211  2222  3321  4321  33311  33321  5431   5441
                             3221  4221  4411         43311  33322  5531
                             3311  4311                      33331  33332
                             4211                            43321  43322
                                                             44311  43331
                                                             53311  44321
                                                                    44411
                                                                    53321
                                                                    54311

References

  • Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.

Links

Crossrefs

Cf. A006918, A096771, A257990, A263297, A325168.
Cf. A325178, A325180, A325181, A325182, A325192, A325197, A325199, A325200.

Programs

  • Mathematica
    durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==2&]],{n,0,30}]

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.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Apr 11 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A325197.

Examples

			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)

Links

Crossrefs

Column k=2 of A325200.
Cf. A046660, A065770, A071724, A243055, A325166, A325169, A325178, A325188, A325189, A325191, A325195, A325197, A325198.

Programs

  • Mathematica
    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}]

A325194 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with co-rank n - k, where co-rank is the greater of the length and the largest part.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 7, 1, 0, 0, 0
Offset: 0

Views

Author

Gus Wiseman, Apr 13 2019

Keywords

Examples

			Triangle begins:
  1
  0  0
  0  1  0
  0  0  0  0
  0  0  2  0  0
  0  0  0  1  0  0
  0  0  0  2  1  0  0
  0  0  0  0  2  0  0  0
  0  0  0  0  2  3  0  0  0
  0  0  0  0  0  2  3  0  0  0
  0  0  0  0  0  2  4  2  0  0  0
  0  0  0  0  0  0  2  5  1  0  0  0
  0  0  0  0  0  0  2  4  7  1  0  0  0
  0  0  0  0  0  0  0  2  6  6  0  0  0  0
  0  0  0  0  0  0  0  2  4  9  7  0  0  0  0
  0  0  0  0  0  0  0  0  2  6 11  5  0  0  0  0
  0  0  0  0  0  0  0  0  2  4 10 14  5  0  0  0  0
  0  0  0  0  0  0  0  0  0  2  6 13 15  3  0  0  0  0
  0  0  0  0  0  0  0  0  0  2  4 10 19 17  2  0  0  0  0
  0  0  0  0  0  0  0  0  0  0  2  6 14 22 17  1  0  0  0  0
  0  0  0  0  0  0  0  0  0  0  2  4 10 21 29 17  1  0  0  0  0
Row n = 16 counts the following partitions:
  (8)         (72)       (64)      (533)    (444)
  (11111111)  (711)      (622)     (542)    (3333)
              (2211111)  (631)     (551)    (4332)
              (3111111)  (6211)    (5222)   (4422)
                         (61111)   (5321)   (4431)
                         (222211)  (5411)
                         (322111)  (32222)
                         (331111)  (33221)
                         (421111)  (33311)
                         (511111)  (42221)
                                   (43211)
                                   (44111)
                                   (52211)
                                   (53111)

Crossrefs

Column sums are A000041. Row sums are A325193.
Cf. A051924, A071724, A096771, A115720, A263297, A325178, A325192.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[k],Max[Length[#],Max[#]]==n-k&]],{n,0,10},{k,0,n}]
Previous Showing 11-14 of 14 results.

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