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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A362051 Number of integer partitions of 2n without a nonempty initial consecutive subsequence summing to n.

Original entry on oeis.org

1, 1, 2, 6, 11, 27, 44, 93, 149, 271, 432, 744, 1109, 1849, 2764, 4287, 6328, 9673, 13853, 20717, 29343, 42609, 60100, 85893, 118475, 167453, 230080, 318654, 433763, 595921, 800878, 1090189, 1456095, 1957032, 2600199, 3465459, 4558785, 6041381, 7908681
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

Even bisection of A362558.
a(0) = 1; a(n) = A000041(2n) - A322439(n). - Alois P. Heinz, Apr 27 2023

Examples

			The a(1) = 1 through a(4) = 11 partitions:
  (2)  (4)   (6)     (8)
       (31)  (42)    (53)
             (51)    (62)
             (222)   (71)
             (411)   (332)
             (2211)  (521)
                     (611)
                     (3221)
                     (3311)
                     (5111)
                     (32111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(4).

Crossrefs

The version for compositions is A000302, bisection of A213173.
The complement is counted by A322439.
Even bisection of A362558.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with all equal run-sums.
A325347 counts partitions with integer median, complement A307683.
A353836 counts partitions by number of distinct run-sums.
A359893/A359901/A359902 count partitions by median.
Cf. A108917, A169942, A237363, A325676, A353864, A360254, A360672, A360675, A360686, A360952, A362560.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],!MemberQ[Accumulate[#],n]&]],{n,0,15}]

A360682 Number of integer partitions of n of length > 2 whose second differences have median 0.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 5, 4, 10, 13, 18, 23, 44, 44, 72, 98, 132, 162, 241, 277, 394, 497, 643, 800, 1076, 1287, 1660, 2078, 2604, 3192, 4065, 4892, 6113, 7490, 9166, 11110, 13717, 16429, 20033, 24201, 29143, 34945, 42251, 50219, 60253, 71852, 85503, 101501, 120899
Offset: 0

Views

Author

Gus Wiseman, Feb 19 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).

Examples

			The a(3) = 1 through a(9) = 13 partitions:
  (111)  (1111)  (11111)  (222)     (22111)    (2222)      (333)
                          (321)     (31111)    (3221)      (432)
                          (2211)    (211111)   (3311)      (531)
                          (21111)   (1111111)  (22211)     (22221)
                          (111111)             (32111)     (33111)
                                               (41111)     (51111)
                                               (221111)    (222111)
                                               (311111)    (321111)
                                               (2111111)   (411111)
                                               (11111111)  (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

Crossrefs

For first differences we have A237363.
For sum instead of median we have A360683.
For mean instead of median we have A360683 - A008619.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions with integer median, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives median of prime indices (times two).
Cf. A000975, A027193, A067538, A114638, A307683, A359908, A360245, A360254, A360687, A360688.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Median[Differences[#,2]]==0&]],{n,0,30}]

A361655 Number of even-length integer partitions of 2n with integer mean.

Original entry on oeis.org

0, 1, 3, 4, 10, 6, 33, 8, 65, 68, 117, 12, 583, 14, 319, 1078, 1416, 18, 3341, 20, 8035, 5799, 1657, 24, 36708, 16954, 3496, 24553, 68528, 30, 192180, 32, 178802, 91561, 14625, 485598, 955142, 38, 29223, 316085, 2622697, 42, 3528870, 44, 2443527, 5740043
Offset: 0

Views

Author

Gus Wiseman, Mar 23 2023

Keywords

Examples

			The a(0) = 0 through a(5) = 6 partitions:
  .  (11)  (22)    (33)      (44)        (55)
           (31)    (42)      (53)        (64)
           (1111)  (51)      (62)        (73)
                   (111111)  (71)        (82)
                             (2222)      (91)
                             (3221)      (1111111111)
                             (3311)
                             (4211)
                             (5111)
                             (11111111)
For example, the partition (4,2,1,1) has length 4 and mean 2, so is counted under a(4).

Links

Crossrefs

Even-length partitions are counted by A027187, bisection A236913.
Including odd-length partitions gives A067538 bisected, ranks A316413.
For median instead of mean we have A361653.
The odd-length version is counted by A361656.
A000041 counts integer partitions, strict A000009.
A051293 counts subsets with integer mean, median A000975.
A058398 counts partitions by mean, see also A008284, A327482.
A325347 counts partitions with integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
A326622 counts factorizations with integer mean.
Cf. A000016, A027193, A067659, A082550, A102627, A237984, A240219, A327475, A348551, A359893.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n], EvenQ[Length[#]]&&IntegerQ[Mean[#]]&]],{n,0,15}]
  • PARI
    a(n)=if(n==0, 0, sumdiv(n, d, polcoef(1/prod(k=1, 2*d, 1 - x^k + O(x*x^(2*(n-d)))), 2*(n-d)))) \\ Andrew Howroyd, Mar 24 2023

Extensions

Terms a(36) and beyond from Andrew Howroyd, Mar 24 2023

A361800 Number of integer partitions of n with the same length as median.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 1, 2, 3, 3, 3, 3, 4, 6, 9, 13, 14, 15, 18, 21, 27, 32, 40, 46, 55, 62, 72, 82, 95, 111, 131, 157, 186, 225, 264, 316, 366, 430, 495, 578, 663, 768, 880, 1011, 1151, 1316, 1489, 1690, 1910, 2158, 2432, 2751, 3100, 3505, 3964, 4486, 5079, 5764
Offset: 1

Views

Author

Gus Wiseman, Apr 07 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).

Examples

			The a(1) = 1 through a(15) = 9 partitions (A=10, B=11):
  1  .  .  22  .  .  331  332  333  433  533  633  733   833   933
           31             431  432  532  632  732  832   932   A32
                               531  631  731  831  931   A31   B31
                                                   4441  4442  4443
                                                         5441  5442
                                                         5531  5532
                                                               6441
                                                               6531
                                                               6621

Crossrefs

For minimum instead of median we have A006141, for twice minimum A237757.
For maximum instead of median we have A047993, for twice length A237753.
For maximum instead of length we have A053263, for twice median A361849.
For mean instead of median we have A206240 (zeros removed).
For minimum instead of length we have A361860.
For twice median we have A362049, ranks A362050.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices.
Cf. A008284, A013580, A027193, A079309, A240219, A362048.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[#]==Median[#]&]],{n,30}]

A361850 Number of strict integer partitions of n such that the maximum is twice the median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 2, 1, 3, 3, 4, 2, 5, 4, 7, 8, 10, 6, 11, 11, 15, 16, 21, 18, 25, 23, 28, 32, 40, 40, 51, 51, 58, 60, 73, 75, 93, 97, 113, 123, 139, 141, 164, 175, 199, 217, 248, 263, 301, 320, 356, 383, 426, 450, 511, 551, 613, 664, 737
Offset: 1

Views

Author

Gus Wiseman, Apr 02 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).

Examples

			The a(7) = 1 through a(20) = 4 strict partitions (A..C = 10..12):
  421  .  .  631  632   .  841   842  843   A51    A52    A53   A54   C62
                  5321     6421       7431  7432   8531   8532  C61   9542
                                      7521  64321  8621         9541  9632
                                                   65321        9631  85421
                                                                9721
The partition (7,4,3,1) has maximum 7 and median 7/2, so is counted under a(15).
The partition (8,6,2,1) has maximum 8 and median 4, so is counted under a(17).

Crossrefs

For minimum instead of median we have A241035, non-strict A237824.
For length instead of median we have A241087, non-strict A237755.
The non-strict version is A361849, ranks A361856.
The non-strict complement is counted by A361857, ranks A361867.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A359907 counts strict partitions with integer median
A360005 gives median of prime indices (times two), distinct A360457.
Cf. A027193, A067659, A079309, A111907, A116608, A359897, A359908, A360952, A361851, A361858, A361859, A361860.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Max@@#==2*Median[#]&]],{n,30}]

A360690 Number of integer partitions of n with non-integer median of multiplicities.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 5, 6, 8, 8, 14, 12, 21, 20, 31, 36, 57, 61, 94, 108, 157, 188, 261, 305, 409, 484, 632, 721, 942, 1083, 1376, 1585, 2004, 2302, 2860, 3304, 4103, 4742, 5849, 6745, 8281, 9599, 11706, 13605, 16481, 19176, 23078, 26838, 32145, 37387, 44465
Offset: 1

Views

Author

Gus Wiseman, Feb 22 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).

Examples

			The a(1) = 0 through a(9) = 8 partitions:
  .  .  .  (211)  (221)  (411)    (322)    (332)      (441)
                  (311)  (21111)  (331)    (422)      (522)
                                  (511)    (611)      (711)
                                  (22111)  (22211)    (22221)
                                  (31111)  (41111)    (33111)
                                           (2111111)  (51111)
                                                      (2211111)
                                                      (3111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is not counted under a(8).

Crossrefs

These partitions have ranks A360554.
The complement is counted by A360687, ranks A360553.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360069 = partitions with integer mean of multiplicities, ranks A067340.
Cf. A000041, A000975, A090794, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

A360691 Number of integer partitions of n with non-integer median of 0-prepended first differences.

Original entry on oeis.org

0, 1, 0, 1, 2, 4, 3, 4, 5, 10, 10, 15, 22, 26, 34, 42, 57, 63, 85, 105, 121, 149, 202, 230, 305, 355, 459, 544, 687, 778, 991, 1130, 1396, 1598, 1947, 2258, 2761, 3143, 3820, 4412, 5330, 6104, 7404, 8499, 10105, 11694, 13922, 15917, 18904, 21646, 25462, 29213
Offset: 1

Views

Author

Gus Wiseman, Feb 22 2023

Keywords

Comments

All of these partitions have even length.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(1) = 0 through a(10) = 10 partitions:
  .  (11)  .  (31)  (32)    (33)    (52)    (53)    (54)      (55)
                    (2111)  (51)    (2221)  (71)    (72)      (73)
                            (2211)  (4111)  (3311)  (3222)    (91)
                            (3111)          (5111)  (6111)    (3322)
                                                    (321111)  (3331)
                                                              (4411)
                                                              (5311)
                                                              (7111)
                                                              (322111)
                                                              (421111)

Crossrefs

For median 0 we have A360254, ranks A360558.
These partitions have ranks A360557, complement A360556.
The complement is counted by A360688.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A237363, A240219, A359907, A359908, A360455, A360555.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Differences[Prepend[Reverse[#],0]]]]&]],{n,30}]

A361391 Number of strict integer partitions of n with non-integer mean.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 4, 2, 4, 5, 11, 0, 17, 15, 13, 15, 37, 18, 53, 24, 48, 78, 103, 23, 111, 152, 143, 123, 255, 110, 339, 238, 372, 495, 377, 243, 759, 845, 873, 414, 1259, 842, 1609, 1383, 1225, 2281, 2589, 1285, 2827, 2518, 3904, 3836, 5119, 3715, 4630
Offset: 0

Views

Author

Gus Wiseman, Mar 11 2023

Keywords

Comments

Are 1, 2, 4, 6, 12 the only zeros?

Examples

			The a(3) = 1 through a(11) = 11 partitions:
  {2,1}  .  {3,2}  .  {4,3}    {4,3,1}  {5,4}  {5,3,2}    {6,5}
            {4,1}     {5,2}    {5,2,1}  {6,3}  {5,4,1}    {7,4}
                      {6,1}             {7,2}  {6,3,1}    {8,3}
                      {4,2,1}           {8,1}  {7,2,1}    {9,2}
                                               {4,3,2,1}  {10,1}
                                                          {5,4,2}
                                                          {6,3,2}
                                                          {6,4,1}
                                                          {7,3,1}
                                                          {8,2,1}
                                                          {5,3,2,1}

Crossrefs

The strict complement is counted by A102627.
The non-strict version is ranked by A348551, complement A316413.
The non-strict version is counted by A349156, complement A067538.
For median instead of mean we have A360952, complement A359907.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A307683 counts partitions with non-integer median, ranks A359912.
A325347 counts partitions with integer median, ranks A359908.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement of A237984.
A327475 counts subsets with integer mean.
Cf. A051293, A082550, A143773, A175397, A175761, A240219, A240850, A326027, A326641, A326849, A359897.

Programs

  • Maple
    a:= proc(m) option remember; local b; b:=
      proc(n, i, t) option remember; `if`(i*(i+1)/2Alois P. Heinz, Mar 16 2023
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!IntegerQ[Mean[#]]&]],{n,0,30}]

Extensions

a(31)-a(55) from Alois P. Heinz, Mar 16 2023

A361653 Number of even-length integer partitions of n with integer median.

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 5, 3, 11, 7, 17, 16, 32, 31, 52, 55, 90, 99, 144, 167, 236, 273, 371, 442, 587, 696, 901, 1078, 1379, 1651, 2074, 2489, 3102, 3707, 4571, 5467, 6692, 7982, 9696, 11543, 13949, 16563, 19891, 23572, 28185, 33299, 39640, 46737, 55418, 65164
Offset: 0

Views

Author

Gus Wiseman, Mar 23 2023

Keywords

Comments

The median of an even-length multiset is the average of the two middle parts.
Because any odd-length partition has integer median, the odd-length version is counted by A027193, strict case A067659.

Examples

			The a(2) = 1 through a(9) = 7 partitions:
  (11)  .  (22)    (2111)  (33)      (2221)    (44)        (3222)
           (31)            (42)      (4111)    (53)        (4221)
           (1111)          (51)      (211111)  (62)        (4311)
                           (3111)              (71)        (6111)
                           (111111)            (2222)      (321111)
                                               (3221)      (411111)
                                               (3311)      (21111111)
                                               (5111)
                                               (221111)
                                               (311111)
                                               (11111111)
For example, the partition (4,3,1,1) has length 4 and median 2, so is counted under a(9).

Crossrefs

The odd-length version is counted by A027193, strict A067659.
Including odd-length partitions gives A307683, complement A325347.
For mean instead of median we have A361655, any length A067538.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median, mean A051293.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A008284, A013580, A079309, A240219, A240850, A349156, A359897, A359908, A359912, A360005, A360952.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&IntegerQ[Median[#]]&]],{n,0,30}]

A361656 Number of odd-length integer partitions of n with integer mean.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 4, 2, 1, 9, 8, 2, 13, 2, 16, 51, 1, 2, 58, 2, 85, 144, 57, 2, 49, 194, 102, 381, 437, 2, 629, 2, 1, 956, 298, 2043, 1954, 2, 491, 2293, 1116, 2, 4479, 2, 6752, 14671, 1256, 2, 193, 8035, 4570, 11614, 22143, 2, 28585, 39810, 16476, 24691, 4566
Offset: 0

Views

Author

Gus Wiseman, Mar 24 2023

Keywords

Comments

These are partitions of n whose length is an odd divisor of n.

Examples

			The a(1) = 1 through a(10) = 8 partitions (A = 10):
  1   2   3     4   5       6     7         8   9           A
          111       11111   222   1111111       333         22222
                            321                 432         32221
                            411                 441         33211
                                                522         42211
                                                531         43111
                                                621         52111
                                                711         61111
                                                111111111
For example, the partition (3,3,2,1,1) has length 5 and mean 2, so is counted under a(10).

Links

Crossrefs

Odd-length partitions are counted by A027193, bisection A236559.
Including even-length gives A067538 bisected, strict A102627, ranks A316413.
The even-length version is counted by A361655.
A000041 counts integer partitions, strict A000009.
A027187 counts even-length partitions, bisection A236913.
A051293 counts subsets with integer mean, median A000975.
A058398 counts partitions by mean, see also A008284, A327482.
A325347 counts partitions with integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
A326622 counts factorizations with integer mean.
Cf. A000016, A067659, A082550, A237984, A240219, A327475, A348551, A361653.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&IntegerQ[Mean[#]]&]],{n,0,30}]
  • PARI
    a(n)=if(n==0, 0, sumdiv(n, d, if(d%2, polcoef(1/prod(k=1, d, 1 - x^k + O(x^(n-d+1))), n-d)))) \\ Andrew Howroyd, Mar 24 2023
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