A361849 -id:A361849 - cp's OEIS frontend

A362049 Number of integer partitions of n such that (length) = 2*(median).

0, 1, 0, 0, 0, 0, 1, 3, 3, 3, 3, 3, 3, 4, 5, 9, 12, 19, 22, 29, 32, 39, 43, 51, 57, 70, 81, 101, 123, 153, 185, 230, 272, 328, 386, 454, 526, 617, 708, 824, 951, 1106, 1277, 1493, 1727, 2020, 2344, 2733, 3164, 3684, 4245, 4914, 5647, 6502, 7438, 8533, 9730

Offset: 1

Gus Wiseman, Apr 10 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). All of these partitions have even length, because an odd-length multiset cannot have fractional median.

			The a(13) = 3 through a(15) = 5 partitions:
  (7,2,2,2)  (8,2,2,2)      (9,2,2,2)
  (8,2,2,1)  (9,2,2,1)      (10,2,2,1)
  (8,3,1,1)  (9,3,1,1)      (10,3,1,1)
             (3,3,3,3,1,1)  (3,3,3,3,2,1)
                            (4,3,3,3,1,1)

For maximum instead of median we have A237753.
For minimum instead of median we have A237757.
For maximum instead of length we have A361849, ranks A361856.
This is the equal case of A362048.
These partitions have 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, distinct A360457.
Cf. A008284, A013580, A027193, A079309, A237800, A240219, A361801, A361848, A361858, A361859.

Table[Length[Select[IntegerPartitions[n],Length[#]==2*Median[#]&]],{n,30}]

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

Gus Wiseman, Apr 07 2023

nonn

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

			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

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.

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

Gus Wiseman, Apr 02 2023

nonn

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

			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).

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.

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

A363221 Number of strict integer partitions of n such that (length) * (maximum) <= 2n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 19, 23, 26, 29, 37, 39, 49, 55, 62, 71, 84, 93, 108, 118, 141, 149, 188, 193, 217, 257, 279, 318, 369, 376, 441, 495, 572, 587, 692, 760, 811, 960, 1046, 1065, 1307, 1387, 1550, 1703, 1796, 2041, 2295, 2456, 2753, 3014

Offset: 1

Gus Wiseman, May 23 2023

nonn

Also strict partitions such that (maximum) <= 2*(mean).

These are strict partitions whose complement (see A361851) has size <= n.

			The partition y = (4,3,1) has length 3 and maximum 4, and 3*4 <= 2*8, so y is counted under a(8). The complement of y has size 4, which is less than or equal to n = 8.

The equal case for median is A361850, non-strict A361849 (ranks A361856).
The non-strict version is A361851, A361848 for median.
The equal case is A361854, non-strict A361853 (ranks A361855).
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
Cf. A111907, A237984, A240219, A241061, A241086, A324521, A324562, A349156, A360068, A360241, A361394, A361852, A361906.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Max@@#<=2*Mean[#]&]],{n,30}]

cp's OEIS Frontend

A362049 Number of integer partitions of n such that (length) = 2*(median).

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

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

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A363221 Number of strict integer partitions of n such that (length) * (maximum) <= 2n.

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Mathematica