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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A361854 Number of strict integer partitions of n such that (length) * (maximum) = 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 2, 0, 5, 0, 6, 3, 5, 0, 11, 6, 8, 7, 10, 0, 36, 0, 14, 16, 16, 29, 43, 0, 21, 36, 69, 0, 97, 0, 35, 138, 33, 0, 150, 61, 137, 134, 74, 0, 231, 134, 265, 229, 56, 0, 650, 0, 65, 749, 267, 247, 533, 0, 405, 565
Offset: 1

Views

Author

Gus Wiseman, Mar 29 2023

Keywords

Comments

Also strict partitions satisfying (maximum) = 2*(mean).
These are strict partitions where both the diagram and its complement (see example) have size n.

Examples

			The a(n) strict partitions for selected n (A..E = 10..14):
  n=9:  n=12:  n=14:  n=15:  n=16:  n=18:  n=20:  n=21:  n=22:
--------------------------------------------------------------
  621   831    7421   A32    8431   C42    A532   E43    B542
        6321          A41    8521   C51    A541   E52    B632
                                    9432   A631   E61    B641
                                    9531   A721          B731
                                    9621   85421         B821
                                           86321
The a(20) = 6 strict partitions are: (10,7,2,1), (10,6,3,1), (10,5,4,1), (10,5,3,2), (8,6,3,2,1), (8,5,4,2,1).
The strict partition y = (8,5,4,2,1) has diagram:
  o o o o o o o o
  o o o o o . . .
  o o o o . . . .
  o o . . . . . .
  o . . . . . . .
Since the partition and its complement (shown in dots) have the same size, y is counted under a(20).

Crossrefs

For minimum instead of mean we have A241035, non-strict A118096.
For length instead of mean we have A241087, non-strict A237753.
For median instead of mean we have A361850, non-strict A361849.
The non-strict version is A361853.
These partitions have ranks A361855 /\ A005117.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A008289 counts strict partitions by length.
A102627 counts strict partitions with integer mean, non-strict A067538.
A116608 counts partitions by number of distinct parts.
A268192 counts partitions by complement size, ranks A326844.
Cf. A111907, A237755, A240850, A326849 A359897, A360068, A360071, A360243, A361848, A361851, A361852, A361906.

Programs

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

A361861 Number of integer partitions of n where the median is twice the minimum.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 5, 5, 8, 11, 16, 20, 28, 38, 53, 67, 87, 111, 146, 183, 236, 297, 379, 471, 591, 729, 909, 1116, 1376, 1682, 2065, 2507, 3055, 3699, 4482, 5395, 6501, 7790, 9345, 11153, 13316, 15839, 18844, 22333, 26466, 31266, 36924, 43478, 51177
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(4) = 1 through a(11) = 11 partitions:
  (31)  (221)  (321)  (421)   (62)     (621)    (442)     (542)
                      (2221)  (521)    (4221)   (721)     (821)
                              (3221)   (4311)   (5221)    (6221)
                              (3311)   (22221)  (5311)    (6311)
                              (22211)  (32211)  (32221)   (33221)
                                                (33211)   (42221)
                                                (42211)   (43211)
                                                (222211)  (52211)
                                                          (222221)
                                                          (322211)
                                                          (2222111)
The partition (3,2,2,2,1,1) has median 2 and minimum 1, so is counted under a(11).
The partition (5,4,2) has median 4 and minimum 2, so is counted under a(11).

Crossrefs

For maximum instead of median we have A118096.
For length instead of median we have A237757, without the coefficient A006141.
With minimum instead of twice minimum we have A361860.
A000041 counts integer partitions, strict A000009.
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.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A027193, A039900, A053263, A067659, A111907, A116608, A237753, A237755, A237824, A361848, A361853.

Programs

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

A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 15, 14, 32, 34, 65, 55, 150, 100, 225, 237, 425, 296, 824, 489, 1267, 1133, 1809, 1254, 4018, 2142, 4499, 4550, 7939, 4564, 14571, 6841, 18285, 16047, 23408, 17495, 52545, 21636, 49943, 51182, 92516, 44582, 144872, 63260, 175318, 169232, 205353
Offset: 0

Views

Author

Gus Wiseman, May 23 2023

Keywords

Comments

Equivalently, n = (length)*(minimum).

Examples

			The a(2) = 1 through a(7) = 14 partitions:
  (31)  (321)  (62)    (32221)  (93)      (3222221)
        (411)  (3221)  (33211)  (552)     (3322211)
               (3311)  (42211)  (642)     (3332111)
               (4211)  (43111)  (732)     (4222211)
               (5111)  (52111)  (822)     (4322111)
                       (61111)  (322221)  (4331111)
                                (332211)  (4421111)
                                (333111)  (5222111)
                                (422211)  (5321111)
                                (432111)  (5411111)
                                (441111)  (6221111)
                                (522111)  (6311111)
                                (531111)  (7211111)
                                (621111)  (8111111)
                                (711111)

Crossrefs

Removing the factor 2 gives A099777.
Taking maximum instead of mean and including odd indices gives A118096.
For length instead of mean and including odd indices we have A237757.
For (maximum) = 2*(mean) see A361851, A361852, A361853, A361854, A361855.
For median instead of mean we have A361861.
These partitions have ranks A363133.
For maximum instead of minimum we have A363218.
For median instead of minimum we have A363224.
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.
A268192 counts partitions by complement size, ranks A326844.
Cf. A053263, A111907, A237753, A237755, A237824, A327482, A349156, A361906, A361907, A363134.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],2*Min@@#==Mean[#]&]],{n,0,15}]
  • Python
    from sympy.utilities.iterables import partitions
def A363132(n): return sum(1 for s,p in partitions(n<<1,m=n,size=True) if n==s*min(p,default=0)) if n else 0 # Chai Wah Wu, Sep 21 2023

Extensions

a(31)-a(46) from Chai Wah Wu, Sep 21 2023

A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 3, 2, 2, 0, 7, 0, 3, 6, 10, 0, 13, 0, 17, 10, 5, 0, 40, 12, 6, 18, 34, 0, 62, 0, 50, 24, 8, 60, 125, 0, 9, 32, 169, 0, 165, 0, 95, 176, 11, 0, 373, 114, 198, 54, 143, 0, 384, 254, 574, 66, 14, 0, 1090, 0, 15, 748, 633, 448, 782, 0, 286
Offset: 1

Views

Author

Gus Wiseman, Apr 10 2023

Keywords

Comments

In terms of partition diagrams, these are partitions whose rectangle from the left (length times minimum) has the same size as the complement.

Examples

			The a(4) = 1 through a(12) = 7 partitions:
  (31)  .  (321)  .  (62)    (441)  (32221)  .  (93)
                     (3221)  (522)  (33211)     (642)
                     (3311)                     (4431)
                                                (5322)
                                                (322221)
                                                (332211)
                                                (333111)
The partition y = (4,4,3,1) has maximum 4 and minimum 1 and mean 3, and 4 - 1 = 3, so y is counted under a(12). The diagram of y is:
  o o o o
  o o o o
  o o o .
  o . . .
Both the rectangle from the left and the complement have size 4.

Crossrefs

Positions of zeros are 1 and A000040.
For length instead of mean we have A237832.
For minimum instead of mean we have A118096.
These partitions have ranks A362047.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A067538 counts partitions with integer mean.
A097364 counts partitions by (maximum) - (minimum).
A243055 subtracts the least prime index from the greatest.
A326844 gives the diagram complement size of Heinz partition.
Cf. A237984, A240219, A326836, A326837, A327482, A237755, A237824, A349156, A359360, A360068, A360241, A361853.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Max@@#-Min@@#==Mean[#]&]],{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

Views

Author

Gus Wiseman, May 23 2023

Keywords

Comments

Also strict partitions such that (maximum) <= 2*(mean).
These are strict partitions whose complement (see A361851) has size <= n.

Examples

			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.

Crossrefs

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.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Max@@#<=2*Mean[#]&]],{n,30}]
Previous Showing 11-15 of 15 results.

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