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.

Showing 1-4 of 4 results.

A325413 Largest sum of the omega-sequence of an integer partition of n.

Original entry on oeis.org

0, 1, 3, 5, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2019

Keywords

Comments

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1) with sum 13.
Appears to contain all nonnegative integers except 2, 4, 6, 7, and 11.

Examples

			The partitions of 9 organized by sum of omega-sequence (first column) are:
   1: (9)
   4: (333)
   5: (81) (72) (63) (54)
   7: (621) (531) (432)
   8: (711) (522) (441)
   9: (6111) (3222) (222111)
  10: (51111) (33111) (22221) (111111111)
  11: (411111)
  12: (5211) (4311) (4221) (3321) (3111111) (2211111)
  13: (42111) (32211) (21111111)
  14: (321111)
The largest term in the first column is 14, so a(9) = 14.

Crossrefs

Row lengths of A325414.
Cf. A181819, A225486, A323014, A323023, A325238, A325249, A325412, A325415, A325416.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325414 (omega-sequence sum).

Programs

  • Mathematica
    omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Max[Total/@omseq/@IntegerPartitions[n]],{n,0,30}]

A325415 Number of distinct sums of omega-sequences of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 10, 11, 13, 12, 15, 14, 16, 18, 18, 18, 21, 20, 23, 23, 24, 24, 27, 27, 28, 29, 30, 30, 34, 32, 34, 35, 36, 37, 39, 38, 40, 41, 43, 42, 45, 44, 46, 48, 48, 48, 51, 50, 53, 53, 54, 54, 57, 57, 58, 59, 60, 60, 64
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2019

Keywords

Comments

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1) with sum 13.

Examples

			The partitions of 9 organized by sum of omega sequence (first column) are:
   1: (9)
   4: (333)
   5: (81) (72) (63) (54)
   7: (621) (531) (432)
   8: (711) (522) (441)
   9: (6111) (3222) (222111)
  10: (51111) (33111) (22221) (111111111)
  11: (411111)
  12: (5211) (4311) (4221) (3321) (3111111) (2211111)
  13: (42111) (32211) (21111111)
  14: (321111)
There are a total of 11 distinct sums {1,4,5,7,8,9,10,11,12,13,14}, so a(9) = 11.

Crossrefs

Number of nonzero terms in row n of A325414.
Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325249, A325277, A325412, A325413, A325416.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325414 (omega-sequence sum).

Programs

  • Mathematica
    omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Length[Union[Total/@omseq/@IntegerPartitions[n]]],{n,0,30}]

A325412 Number of distinct omega-sequences of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 5, 10, 9, 14, 15, 20, 21, 33, 30, 39, 45, 54, 54, 69, 68, 85, 90, 100, 104, 128, 127, 141, 153, 172, 175, 205, 203, 229, 240, 257, 274, 308, 309, 335, 356, 390, 395, 437, 444, 481, 506, 530, 549, 602, 609, 648, 672, 710, 727, 777, 798, 848, 871
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2019

Keywords

Comments

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1).

Examples

			The a(1) = 1 through a(9) = 15 omega-sequences:
  (1)  (1)   (1)    (1)     (1)     (1)     (1)      (1)      (1)
       (21)  (31)   (21)    (51)    (21)    (71)     (21)     (31)
             (221)  (41)    (221)   (31)    (221)    (41)     (91)
                    (221)   (3221)  (61)    (331)    (81)     (221)
                    (3221)  (4221)  (221)   (3221)   (221)    (331)
                                    (331)   (4221)   (331)    (621)
                                    (421)   (5221)   (421)    (3221)
                                    (3221)  (6221)   (3221)   (4221)
                                    (4221)  (43221)  (4221)   (5221)
                                    (5221)           (5221)   (6221)
                                                     (6221)   (7221)
                                                     (7221)   (8221)
                                                     (43221)  (43221)
                                                     (53221)  (53221)
                                                              (63221)

Crossrefs

Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325277, A325413, A325415.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325414 (omega-sequence sum).

Programs

  • Mathematica
    omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Length[Union[omseq/@IntegerPartitions[n]]],{n,0,30}]

A325285 Number of integer partitions of n whose omega-sequence has repeated parts.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 13, 17, 26, 36, 54, 66, 98, 125, 164, 214, 285, 354, 468, 585, 745, 945, 1195, 1477, 1864, 2317, 2867, 3544, 4383, 5348, 6589, 8028, 9778, 11885, 14403, 17362, 20992, 25212, 30239, 36158, 43242, 51408, 61240, 72568, 85989, 101607, 120027
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2019

Keywords

Comments

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1), which has repeated parts, so (32211) is counted under a(9).
The Heinz numbers of these partitions are given by A325411.

Examples

			The a(3) = 1 through a(8) = 17 partitions:
  (21)  (31)   (32)    (42)     (43)      (53)
        (211)  (41)    (51)     (52)      (62)
               (221)   (321)    (61)      (71)
               (311)   (411)    (322)     (332)
               (2111)  (3111)   (331)     (422)
                       (21111)  (421)     (431)
                                (511)     (521)
                                (2221)    (611)
                                (3211)    (3221)
                                (4111)    (4211)
                                (22111)   (5111)
                                (31111)   (22211)
                                (211111)  (32111)
                                          (41111)
                                          (221111)
                                          (311111)
                                          (2111111)

Crossrefs

Cf. A047966, A181819, A323014, A323023, A325247, A325250, A325260, A325262, A325411.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325414 (omega-sequence sum).

Programs

  • Mathematica
    omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Length[Select[IntegerPartitions[n],!UnsameQ@@omseq[#]&]],{n,0,30}]
Showing 1-4 of 4 results.

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