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 1-3 of 3 results.

A363264 Number of integer partitions of n covering an initial interval of positive integers with a more than one co-mode.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 2, 3, 3, 2, 7, 5, 9, 12, 13, 13, 22, 19, 29, 33, 39, 43, 63, 63, 82, 94, 111, 119, 159, 164, 203, 229, 272, 301, 370, 400, 479, 538, 628, 692, 826, 904, 1053, 1181, 1353, 1502, 1742, 1919, 2205, 2456, 2790, 3097, 3539, 3911, 4435, 4929
Offset: 0

Views

Author

Gus Wiseman, Jun 06 2023

Keywords

Comments

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

Crossrefs

For parts instead of multiplicities we have A000009, complement A087897.
For multisets instead of partitions we have A363224, complement A105039.
The complement is counted by A363263.
For mode we have A363485, complement A363484.
A000041 counts integer partitions, A000009 covering an initial interval.
A067029 counts minima in prime factorization, co-modes A362613.
A071178 counts maxima in prime factorization, modes A362611.
A097979 counts normal multisets with a unique mode, complement A363262.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A096765, A117989, A243737, A275870, A362612.

Programs

  • Mathematica
    comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]>1&]]],{n,0,30}]

A363484 Number of integer partitions of n covering an initial interval of positive integers with a unique mode.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 5, 6, 6, 8, 11, 12, 17, 20, 21, 27, 35, 38, 50, 56, 65, 76, 95, 105, 125, 146, 167, 198, 233, 252, 305, 351, 394, 457, 522, 585, 681, 778, 878, 994, 1135, 1269, 1446, 1638, 1828, 2067, 2339, 2613, 2940, 3301, 3684, 4143, 4634, 5156, 5771
Offset: 0

Views

Author

Gus Wiseman, Jun 05 2023

Keywords

Comments

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

Examples

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (11)  (111)  (211)   (221)    (21111)   (2221)     (3221)
                    (1111)  (2111)   (111111)  (3211)     (22211)
                            (11111)            (22111)    (32111)
                                               (211111)   (221111)
                                               (1111111)  (2111111)
                                                          (11111111)

Crossrefs

For parts instead of multiplicities we have A096765, complement A025147.
For multisets instead of partitions we have A097979, complement A363262.
For co-mode we have A363263, complement A363264.
The complement is counted by A363485.
A000041 counts integer partitions, A000009 covering an initial interval.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A105039, A117989, A243737, A362612.

Programs

  • Mathematica
    Table[If[n==0,0,Length[Select[IntegerPartitions[n], Union[#]==Range[Max@@#]&&Length[Commonest[#]]==1&]]],{n,0,30}]

A363485 Number of integer partitions of n covering an initial interval of positive integers with more than one mode.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 1, 3, 1, 2, 6, 5, 3, 8, 4, 8, 11, 13, 9, 17, 17, 19, 25, 24, 23, 44, 35, 39, 54, 55, 63, 83, 79, 86, 104, 119, 125, 157, 164, 178, 220, 237, 251, 297, 324, 357, 413, 439, 486, 562, 607, 673, 765, 828, 901, 1040, 1117, 1220
Offset: 0

Views

Author

Gus Wiseman, Jun 06 2023

Keywords

Comments

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

Examples

			The a(n) partitions for n = {3, 6, 12, 15, 16, 18}:
  (21)  (321)   (332211)    (54321)       (443221)    (4433211)
        (2211)  (3222111)   (433221)      (3332221)   (5432211)
                (22221111)  (443211)      (4332211)   (43332111)
                            (33222111)    (33322111)  (333222111)
                            (322221111)   (43222111)  (333321111)
                            (2222211111)              (3322221111)
                                                      (32222211111)
                                                      (222222111111)

Crossrefs

For parts instead of multiplicities we have A025147, complement A096765.
For co-mode we have A363264, complement A363263.
The complement is counted by A363484.
A000041 counts integer partitions, A000009 covering an initial interval.
A071178 counts maxima in prime factorization, modes A362611.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A105039, A117989, A243737, A362612.

Programs

  • Mathematica
    Table[If[n==0,0,Length[Select[IntegerPartitions[n], Union[#]==Range[Max@@#]&&Length[Commonest[#]]>1&]]],{n,0,30}]
Showing 1-3 of 3 results.

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