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 1-5 of 5 results.

A363723 Number of integer partitions of n having a unique mode equal to the mean, i.e., partitions whose mean appears more times than each of the other parts.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 60, 15, 18, 37, 60, 2, 129, 2, 104, 80, 35, 104, 352, 2, 49, 168, 501, 2, 556, 2, 489, 763, 92, 2, 1799, 292, 985, 649, 1296, 2, 2233, 1681, 3379, 1204, 225, 2, 10661
Offset: 0

Views

Author

Gus Wiseman, Jun 24 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 in {a,a,b,b,b,c,d,d,d} are {b,d}.

Examples

			The a(n) partitions for n = 6, 8, 12, 14, 16 (A..G = 10..16):
  (6)       (8)         (C)             (E)               (G)
  (33)      (44)        (66)            (77)              (88)
  (222)     (2222)      (444)           (2222222)         (4444)
  (111111)  (3221)      (3333)          (3222221)         (5443)
            (11111111)  (4332)          (3322211)         (6442)
                        (5331)          (4222211)         (7441)
                        (222222)        (11111111111111)  (22222222)
                        (322221)                          (32222221)
                        (422211)                          (33222211)
                        (111111111111)                    (42222211)
                                                          (52222111)
                                                          (1111111111111111)

Crossrefs

Partitions containing their mean are counted by A237984, ranks A327473.
For median instead of mode we have A240219, ranks A359889.
Partitions missing their mean are counted by A327472, ranks A327476.
The case of non-constant partitions is A362562.
Including median also gives A363719, ranks A363727.
Allowing multiple modes gives A363724.
Requiring multiple modes gives A363731.
For median instead of mean we have A363740.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median.
A362608 counts partitions with a unique mode.
Cf. A325347, A326567/A326568, A363720, A363725, A363730.

Programs

  • Mathematica
    modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],{Mean[#]}==modes[#]&]],{n,30}]

A363724 Number of integer partitions of n whose mean is a mode, i.e., partitions whose mean appears at least as many times as each of the other parts.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 5, 6, 2, 15, 2, 8, 15, 17, 2, 30, 2, 43, 30, 15, 2, 112, 36, 21, 60, 119, 2, 251, 2, 201, 126, 41, 271, 655, 2, 57, 250, 1060, 2, 1099, 2, 844, 1508, 107, 2, 3484, 802, 2068, 900, 2136, 2, 4558, 3513, 7071, 1630, 259, 2, 20260
Offset: 1

Views

Author

Gus Wiseman, Jun 24 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 in {a,a,b,b,b,c,d,d,d} are {b,d}.

Examples

			The a(n) partitions for n = 6, 10, 12:
  (6)            (10)                   (12)
  (3,3)          (5,5)                  (6,6)
  (2,2,2)        (2,2,2,2,2)            (4,4,4)
  (3,2,1)        (3,2,2,2,1)            (5,4,3)
  (1,1,1,1,1,1)  (4,2,2,1,1)            (6,4,2)
                 (1,1,1,1,1,1,1,1,1,1)  (7,4,1)
                                        (3,3,3,3)
                                        (4,3,3,2)
                                        (5,3,3,1)
                                        (6,3,2,1)
                                        (2,2,2,2,2,2)
                                        (3,2,2,2,2,1)
                                        (3,3,2,2,1,1)
                                        (4,2,2,2,1,1)
                                        (1,1,1,1,1,1,1,1,1,1,1,1)

Crossrefs

For parts instead of modes we have A237984, complement A327472.
The case of a unique mode is A363723, non-constant A362562.
The case of more than one mode is A363731.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A362608 counts partitions with a unique mode.
A363719 = all three averages equal, ranks A363727, non-constant A363728.
A363720 = all three averages different, ranks A363730, unique mode A363725.
Cf. A240219, A326567/A326568, A363726, A363740.

Programs

  • Mathematica
    modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],MemberQ[modes[#],Mean[#]]&]],{n,30}]

A363740 Number of integer partitions of n whose median appears more times than any other part, i.e., partitions containing a unique mode equal to the median.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 10, 15, 18, 26, 35, 46, 61, 82, 102, 136, 174, 224, 283, 360, 449, 569, 708, 883, 1089, 1352, 1659, 2042, 2492, 3039, 3695, 4492, 5426, 6555, 7889, 9482, 11360, 13602, 16231, 19348, 23005, 27313, 32364, 38303, 45227, 53341, 62800, 73829
Offset: 1

Views

Author

Gus Wiseman, Jun 26 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

Examples

			The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (3221)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Crossrefs

For mean instead of mode we have A240219, see A359894, A359889, A359895, A359897, A359899.
Including mean also gives A363719, ranks A363727.
For mean instead of median we have A363723, see A363724, A363731.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median.
A362608 counts partitions with a unique mode, ranks A356862.
Cf. A027193, A237984, A325347, A362562, A363720, A363725, A363726.

Programs

  • Mathematica
    modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],{Median[#]}==modes[#]&]],{n,30}]

A363731 Number of integer partitions of n whose mean is a mode but not the only mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 5, 0, 1, 8, 5, 0, 12, 0, 19, 14, 2, 0, 52, 21, 3, 23, 59, 0, 122, 0, 97, 46, 6, 167, 303, 0, 8, 82, 559, 0, 543, 0, 355, 745, 15, 0, 1685, 510, 1083, 251, 840, 0, 2325, 1832, 3692, 426, 34, 0, 9599
Offset: 0

Views

Author

Gus Wiseman, Jun 24 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 in {a,a,b,b,b,c,d,d,d} are {b,d}.

Examples

			The a(n) partitions for n = 6, 9, 12, 15, 18:
  (3,2,1)  (4,3,2)  (5,4,3)        (6,5,4)      (7,6,5)
           (5,3,1)  (6,4,2)        (7,5,3)      (8,6,4)
                    (7,4,1)        (8,5,2)      (9,6,3)
                    (6,3,2,1)      (9,5,1)      (10,6,2)
                    (3,3,2,2,1,1)  (4,4,3,3,1)  (11,6,1)
                                   (5,3,3,2,2)  (4,4,3,3,2,2)
                                   (5,4,3,2,1)  (5,5,3,3,1,1)
                                   (7,3,3,1,1)  (6,4,3,3,1,1)
                                                (7,3,3,2,2,1)
                                                (8,3,3,2,1,1)
                                                (3,3,3,2,2,2,1,1,1)
                                                (6,2,2,2,2,1,1,1,1)

Crossrefs

For a unique mode we have A363723, non-constant A362562.
For any number of modes we have A363724.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A237984 counts partitions containing their mean, ranks A327473.
A327472 counts partitions not containing their mean, ranks A327476.
A362608 counts partitions with a unique mode, ranks A356862.
A363719 counts partitions with all three averages equal, ranks A363727.
Cf. A240219, A326567/A326568, A359893, A363720, A363725, A363740.

Programs

  • Mathematica
    modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],MemberQ[modes[#],Mean[#]]&&!{Mean[#]}==modes[#]&]],{n,30}]

A364062 Number of integer partitions of n with unique co-mode 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 6, 2, 8, 6, 9, 6, 16, 7, 21, 12, 23, 18, 39, 17, 47, 32, 59, 40, 86, 44, 110, 72, 131, 95, 188, 103, 233, 166, 288, 201, 389, 244, 490, 347, 587, 440, 794, 524, 974, 727, 1187, 903, 1547, 1106, 1908, 1459, 2303, 1826, 2979, 2198
Offset: 0

Views

Author

Gus Wiseman, Jul 12 2023

Keywords

Comments

These are partitions with at least one 1 but with fewer 1's than each of the other parts.
We define a co-mode in a multiset to be an element that appears at most as many times as each of the other elements. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

Examples

			The a(n) partitions for n = 5, 7, 11, 13, 15:
  (221)    (331)      (551)          (661)            (771)
  (11111)  (2221)     (33221)        (4441)           (44331)
           (1111111)  (33311)        (33331)          (55221)
                      (222221)       (44221)          (442221)
                      (2222111)      (332221)         (3322221)
                      (11111111111)  (2222221)        (3333111)
                                     (22222111)       (22222221)
                                     (1111111111111)  (222222111)
                                                      (111111111111111)

Crossrefs

For high (or unique) mode we have A241131, ranks A360013.
For low mode we have A241131, ranks A360015.
Allowing any unique co-mode gives A362610, ranks A359178.
These partitions have ranks A364061.
Adding all 1-free partitions gives A364159, ranks A364158.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A237984 counts partitions containing their mean, ranks A327473.
A327472 counts partitions not containing their mean, ranks A327476.
A362608 counts partitions w/ unique mode, ranks A356862, complement A362605.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Cf. A002865, A027336, A325347, A098859, A360014, A362562, A362606, A363720, A363723, A363725, A363726.

Programs

  • Mathematica
    comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],comodes[#]=={1}&]],{n,0,30}]
Showing 1-5 of 5 results.

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