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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A326842 Number of integer partitions of n whose parts all divide n and whose length also divides n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 21, 2, 5, 6, 9, 2, 22, 2, 21, 6, 5, 2, 134, 3, 5, 6, 23, 2, 157, 2, 27, 6, 5, 6, 478, 2, 5, 6, 208, 2, 224, 2, 31, 63, 5, 2, 1720, 3, 30, 6, 34, 2, 322, 6, 295, 6, 5, 2, 13899, 2, 5, 68, 126, 8, 429, 2, 42, 6, 358, 2, 19959, 2
Offset: 0

Views

Author

Gus Wiseman, Jul 26 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A326847.

Examples

			The a(1) = 1 through a(8) = 5 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (1111)           (222)                (2222)
                                     (321)                (4211)
                                     (111111)             (11111111)
The a(12) = 21 partitions:
  (12)
  (6,6)
  (4,4,4)
  (6,3,3)
  (6,4,2)
  (3,3,3,3)
  (4,3,3,2)
  (4,4,2,2)
  (4,4,3,1)
  (6,2,2,2)
  (6,3,2,1)
  (6,4,1,1)
  (2,2,2,2,2,2)
  (3,2,2,2,2,1)
  (3,3,2,2,1,1)
  (3,3,3,1,1,1)
  (4,2,2,2,1,1)
  (4,3,2,1,1,1)
  (4,4,1,1,1,1)
  (6,2,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)

Links

Crossrefs

Partitions using divisors are A018818.
Partitions whose length divides their sum are A067538.
Cf. A047993, A102627, A316413, A326841, A326843, A326847.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n,All,Divisors[n]],Divisible[n,Length[#]]&]],{n,1,30}]

A359895 Number of odd-length integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 1, 5, 5, 2, 5, 2, 8, 18, 1, 2, 19, 2, 24, 41, 20, 2, 9, 44, 31, 94, 102, 2, 125, 2, 1, 206, 68, 365, 382, 2, 98, 433, 155, 2, 716, 2, 1162, 2332, 196, 2, 17, 1108, 563, 1665, 3287, 2, 3906, 5474, 2005, 3083, 509, 2, 9029
Offset: 0

Views

Author

Gus Wiseman, Jan 20 2023

Keywords

Comments

The length and median of such a partition are integers with product n.

Examples

			The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)  (9)
            (111)       (11111)  (222)  (1111111)       (333)
                                 (321)                  (432)
                                                        (531)
                                                        (111111111)
The a(15) = 18 partitions:
  (15)
  (5,5,5)
  (6,5,4)
  (7,5,3)
  (8,5,2)
  (9,5,1)
  (3,3,3,3,3)
  (4,3,3,3,2)
  (4,4,3,2,2)
  (4,4,3,3,1)
  (5,3,3,2,2)
  (5,3,3,3,1)
  (5,4,3,2,1)
  (5,5,3,1,1)
  (6,3,3,2,1)
  (6,4,3,1,1)
  (7,3,3,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

Links

Crossrefs

This is the odd-length case of A240219, complement A359894, strict A359897.
These partitions are ranked by A359891, complement A359892.
The complement is counted by A359896.
The strict case is A359899, complement A359900.
The version for factorizations is A359910.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A008289, A316313, A327472, A327475, A327482, A359889, A359906.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,0,30}]
  • PARI
    \\ P(n, k, m) is g.f. for k parts of max size m.
P(n, k, m)={polcoef(1/prod(i=1, m, 1 - y*x^i + O(x*x^n)), k, y)}
a(n)={if(n==0, 0, sumdiv(n, d, if(d%2, my(m=n/d, h=d\2, r=n-m*(h+1)+h); polcoef(P(r, h, m)*P(r, h, r), r))))} \\ Andrew Howroyd, Jan 21 2023

Formula

a(p) = 2 for prime p. - Andrew Howroyd, Jan 21 2023

A360254 Number of integer partitions of n with more adjacent equal parts than distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 4, 7, 10, 12, 18, 28, 36, 52, 68, 92, 119, 161, 204, 269, 355, 452, 571, 738, 921, 1167, 1457, 1829, 2270, 2834, 3483, 4314, 5300, 6502, 7932, 9665, 11735, 14263, 17227, 20807, 25042, 30137, 36099, 43264, 51646, 61608, 73291, 87146, 103296
Offset: 0

Views

Author

Gus Wiseman, Feb 20 2023

Keywords

Comments

None of these partitions is strict.
Also the number of integer partitions of n which, after appending 0, have first differences of median 0.

Examples

			The a(3) = 1 through a(9) = 10 partitions:
  (111)  (1111)  (11111)  (222)     (22111)    (2222)      (333)
                          (21111)   (31111)    (22211)     (22221)
                          (111111)  (211111)   (41111)     (33111)
                                    (1111111)  (221111)    (51111)
                                               (311111)    (222111)
                                               (2111111)   (411111)
                                               (11111111)  (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)
For example, the partition y = (4,4,3,1,1,1,1) has 0-appended differences (0,1,2,0,0,0,0), with median 0, so y is counted under a(15).

Crossrefs

The non-prepended version is A237363.
These partitions have ranks A360558.
For any integer median (not just 0) we have A360688.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A067538, A102627, A240219, A359894, A360071, A360244, A360555, A360556.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Length[#]>2*Length[Union[#]]&]],{n,0,30}]

A363946 Triangle read by rows where T(n,k) is the number of integer partitions of n with high mean k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 3, 2, 0, 1, 0, 1, 6, 3, 0, 0, 1, 0, 1, 6, 4, 3, 0, 0, 1, 0, 1, 11, 5, 4, 0, 0, 0, 1, 0, 1, 11, 13, 0, 4, 0, 0, 0, 1, 0, 1, 18, 9, 8, 5, 0, 0, 0, 0, 1, 0, 1, 18, 21, 10, 0, 5, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Gus Wiseman, Jun 30 2023

Keywords

Comments

Extending the terminology of A124944, the "high mean" of a multiset is obtained by taking the mean and rounding up.

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  3  0  1
  0  1  3  2  0  1
  0  1  6  3  0  0  1
  0  1  6  4  3  0  0  1
  0  1 11  5  4  0  0  0  1
  0  1 11 13  0  4  0  0  0  1
  0  1 18  9  8  5  0  0  0  0  1
  0  1 18 21 10  0  5  0  0  0  0  1
  0  1 29 28 12  0  6  0  0  0  0  0  1
  0  1 29 32 18 14  0  6  0  0  0  0  0  1
  0  1 44 43 23 16  0  7  0  0  0  0  0  0  1
  0  1 44 77 27 19  0  0  7  0  0  0  0  0  0  1
Row n = 7 counts the following partitions:
  .  (1111111)  (4111)    (511)  (61)  .  .  (7)
                (3211)    (421)  (52)
                (31111)   (331)  (43)
                (2221)    (322)
                (22111)
                (211111)

Crossrefs

Row sums are A000041.
Column k = 2 is A026905 redoubled, ranks A363950.
For median instead of mean we have triangle A124944, low A124943.
For mode instead of mean we have rank stat A363486, high A363487.
For median instead of mean we have rank statistic A363942, low A363941.
The rank statistic for this triangle is A363944.
The version for low mean is A363945, rank statistic A363943.
For mode instead of mean we have triangle A363953, low A363952.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A002865, A025065, A237984, A327472, A327482, A344296, A362612, A363723, A363724, A363731, A363948.

Programs

  • Mathematica
    meanup[y_]:=If[Length[y]==0,0,Ceiling[Mean[y]]];
Table[Length[Select[IntegerPartitions[n],meanup[#]==k&]],{n,0,15},{k,0,n}]

A360241 Number of integer partitions of n whose distinct parts have integer mean.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 6, 13, 13, 22, 19, 43, 34, 56, 66, 97, 92, 156, 143, 233, 256, 322, 341, 555, 542, 710, 831, 1098, 1131, 1644, 1660, 2275, 2484, 3035, 3492, 4731, 4848, 6063, 6893, 8943, 9378, 12222, 13025, 16520, 18748, 22048, 24405, 31446, 33698, 41558
Offset: 0

Views

Author

Gus Wiseman, Feb 02 2023

Keywords

Examples

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (331)      (44)
                    (31)    (11111)  (42)      (511)      (53)
                    (1111)           (51)      (3211)     (62)
                                     (222)     (31111)    (71)
                                     (321)     (1111111)  (422)
                                     (3111)               (2222)
                                     (111111)             (3221)
                                                          (3311)
                                                          (5111)
                                                          (32111)
                                                          (311111)
                                                          (11111111)
For example, the partition (32111) has distinct parts {1,2,3} with mean 2, so is counted under a(8).

Crossrefs

For parts instead of distinct parts we have A067538, ranked by A316413.
The strict case is A102627.
These partitions are ranked by A326621.
For multiplicities instead of distinct parts: A360069, ranked by A067340.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A051293 counts subsets with integer mean, median A000975.
A058398 counts partitions by mean, also A327482.
A116608 counts partitions by number of distinct parts.
A326619/A326620 gives mean of distinct prime indices.
A326622 counts factorizations with integer mean, strict A328966.
A360071 counts partitions by number of parts and number of distinct parts.
The following count partitions:
- A360242 mean(parts) != mean(distinct parts), ranked by A360246.
- A360243 mean(parts) = mean(distinct parts), ranked by A360247.
- A360250 mean(parts) > mean(distinct parts), ranked by A360252.
- A360251 mean(parts) < mean(distinct parts), ranked by A360253.
Cf. A067539, A078174, A082550, A240219, A316313, A325347, A326567/A326568, A326669, A327475, A349156.

Programs

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

A363945 Triangle read by rows where T(n,k) is the number of integer partitions of n with low mean k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 4, 2, 0, 0, 1, 0, 4, 3, 3, 0, 0, 1, 0, 7, 4, 3, 0, 0, 0, 1, 0, 7, 10, 0, 4, 0, 0, 0, 1, 0, 12, 6, 7, 4, 0, 0, 0, 0, 1, 0, 12, 16, 8, 0, 5, 0, 0, 0, 0, 1, 0, 19, 21, 10, 0, 5, 0, 0, 0, 0
Offset: 0

Views

Author

Gus Wiseman, Jun 30 2023

Keywords

Comments

Extending the terminology of A124943, the "low mean" of a multiset is its mean rounded down.

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  2  0  1
  0  2  2  0  1
  0  4  2  0  0  1
  0  4  3  3  0  0  1
  0  7  4  3  0  0  0  1
  0  7 10  0  4  0  0  0  1
  0 12  6  7  4  0  0  0  0  1
  0 12 16  8  0  5  0  0  0  0  1
  0 19 21 10  0  5  0  0  0  0  0  1
  0 19 24 15 12  0  6  0  0  0  0  0  1
  0 30 32 18 14  0  6  0  0  0  0  0  0  1
  0 30 58 23 16  0  0  7  0  0  0  0  0  0  1
  0 45 47 57  0 19  0  7  0  0  0  0  0  0  0  1
Row k = 8 counts the following partitions:
  .  (41111)     (611)   .  (71)  .  .  .  (8)
     (32111)     (521)      (62)
     (311111)    (5111)     (53)
     (22211)     (431)      (44)
     (221111)    (422)
     (2111111)   (4211)
     (11111111)  (332)
                 (3311)
                 (3221)
                 (2222)

Crossrefs

Row sums are A000041.
Column k = 1 is A025065, ranks A363949.
For median instead of mean we have triangle A124943, high A124944.
Column k = 2 is A363745.
For median instead of mean we have rank statistic A363941, high A363942.
The rank statistic for this triangle is A363943.
The high version is A363946, rank statistic A363944.
For mode instead of mean we have A363952, rank statistic A363486.
For high mode instead of mean we have A363953, rank statistic A363487.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A002865, A026905, A237984, A327472, A327482, A344296, A362612, A363723, A363724, A363731, A363951.

Programs

  • Mathematica
    meandown[y_]:=If[Length[y]==0,0,Floor[Mean[y]]];
Table[Length[Select[IntegerPartitions[n],meandown[#]==k&]],{n,0,15},{k,0,n}]

A361394 Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 8, 11, 15, 20, 30, 38, 49, 65, 83, 108, 139, 178, 224, 286, 358, 437, 550, 684, 837, 1037, 1269, 1553, 1889, 2295, 2770, 3359, 4035, 4843, 5808, 6951, 8312, 9902, 11752, 13958, 16531, 19541, 23037, 27162, 31911, 37488, 43950, 51463, 60127, 70229
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2023

Keywords

Examples

			The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)   (4)    (5)     (6)     (7)
       (11)  (21)  (22)   (32)    (33)    (43)
                   (31)   (41)    (42)    (52)
                   (211)  (221)   (51)    (61)
                          (311)   (321)   (322)
                          (2111)  (411)   (331)
                                  (2211)  (421)
                                  (3111)  (511)
                                          (2221)
                                          (3211)
                                          (4111)

Links

Crossrefs

The complement is counted by A360254, ranks A360558.
These partitions have ranks A361395.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, reverse A058398.
A067538 counts partitions with integer mean, strict A102627.
A116608 counts partitions by number of distinct parts.
Cf. A106529, A144300, A237363, A237832, A240219, A316413, A349156.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>=0, 1, 0),
     `if`(i<1, 0, add(b(n-i*j, i-1, t+`if`(j>0, 2, 0)-j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 19 2023
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],2*Length[Union[#]]>=Length[#]&]],{n,0,30}]

A359899 Number of strict odd-length integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 6, 1, 1, 6, 1, 5, 7, 1, 1, 8, 12, 1, 9, 2, 1, 33, 1, 1, 11, 1, 50, 12, 1, 1, 13, 70, 1, 46, 1, 1, 122, 1, 1, 16, 102, 155, 17, 1, 1, 30, 216, 258, 19, 1, 1, 310, 1, 1, 666, 1, 382, 23, 1, 1, 23, 1596, 1, 393, 1, 1
Offset: 0

Views

Author

Gus Wiseman, Jan 20 2023

Keywords

Examples

			The a(30) = 33 partitions:
  (30)  (11,10,9)  (8,7,6,5,4)
        (12,10,8)  (9,7,6,5,3)
        (13,10,7)  (9,8,6,4,3)
        (14,10,6)  (9,8,6,5,2)
        (15,10,5)  (10,7,6,4,3)
        (16,10,4)  (10,7,6,5,2)
        (17,10,3)  (10,8,6,4,2)
        (18,10,2)  (10,8,6,5,1)
        (19,10,1)  (10,9,6,3,2)
                   (10,9,6,4,1)
                   (11,7,6,4,2)
                   (11,7,6,5,1)
                   (11,8,6,3,2)
                   (11,8,6,4,1)
                   (11,9,6,3,1)
                   (12,7,6,3,2)
                   (12,7,6,4,1)
                   (12,8,6,3,1)
                   (12,9,6,2,1)
                   (13,7,6,3,1)
                   (13,8,6,2,1)
                   (14,7,6,2,1)
                   (11,10,6,2,1)

Links

Crossrefs

Strict odd-length case of A240219, complement A359894, ranked by A359889.
Strict case of A359895, complement A359896, ranked by A359891.
Odd-length case of A359897, complement A359898.
The complement is counted by A359900.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000016, A065795, A066571, A240851, A359903, A359906, A359910.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,0,30}]
  • PARI
    \\ Q(n,k,m) is g.f. for k strict parts of max size m.
Q(n,k,m)={polcoef(prod(i=1, m, 1 + y*x^i + O(x*x^n)), k, y)}
a(n)={if(n==0, 0, sumdiv(n, d, if(d%2, my(m=n/d, h=d\2, r=n-m*(h+1)); if(r>=h*(h+1), polcoef(Q(r, h, m-1)*Q(r, h, r), r)))))} \\ Andrew Howroyd, Jan 21 2023

Formula

a(p) = 1 for prime p. - Andrew Howroyd, Jan 21 2023

A360242 Number of integer partitions of n where the parts do not have the same mean as the distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 11, 19, 25, 43, 49, 82, 103, 136, 183, 258, 314, 435, 524, 687, 892, 1150, 1378, 1788, 2241, 2773, 3399, 4308, 5142, 6501, 7834, 9600, 11726, 14099, 16949, 20876, 25042, 30032, 35732, 43322, 51037, 61650, 72807, 86319, 102983, 122163
Offset: 0

Views

Author

Gus Wiseman, Feb 04 2023

Keywords

Examples

			The a(1) = 0 through a(9) = 19 partitions:
  .  .  .  (211)  (221)   (411)    (322)     (332)      (441)
                  (311)   (3111)   (331)     (422)      (522)
                  (2111)  (21111)  (511)     (611)      (711)
                                   (2221)    (4211)     (3222)
                                   (3211)    (5111)     (3321)
                                   (4111)    (22211)    (4221)
                                   (22111)   (32111)    (4311)
                                   (31111)   (41111)    (5211)
                                   (211111)  (221111)   (6111)
                                             (311111)   (22221)
                                             (2111111)  (32211)
                                                        (33111)
                                                        (42111)
                                                        (51111)
                                                        (321111)
                                                        (411111)
                                                        (2211111)
                                                        (3111111)
                                                        (21111111)
For example, the partition y = (32211) has mean 9/5 and distinct parts {1,2,3} with mean 2, so y is counted under a(9).

Crossrefs

The complement for multiplicities instead of distinct parts is A360068.
The complement is counted by A360243, ranks A360247.
For median instead of mean we have A360244, complement A360245.
These partitions have ranks A360246.
Sum of A360250 and A360251, ranks A360252 and A360253.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A058398 counts partitions by mean, also A327482.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A360071 counts partitions by number of parts and number of distinct parts.
A360241 counts partitions whose distinct parts have integer mean.
Cf. A051293, A067340, A240219, A316313, A326567/A326568, A326619/A326620, A326621, A349156.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Mean[#]!=Mean[Union[#]]&]],{n,0,30}]

A360243 Number of integer partitions of n where the parts have the same mean as the distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 11, 11, 17, 13, 28, 19, 32, 40, 48, 39, 71, 55, 103, 105, 110, 105, 197, 170, 195, 237, 319, 257, 462, 341, 515, 543, 584, 784, 1028, 761, 973, 1153, 1606, 1261, 2137, 1611, 2368, 2815, 2575, 2591, 4393, 3798, 4602, 4663, 5777, 5121
Offset: 0

Views

Author

Gus Wiseman, Feb 04 2023

Keywords

Examples

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (11111)  (51)      (61)       (62)
                                     (222)     (421)      (71)
                                     (321)     (1111111)  (431)
                                     (2211)               (521)
                                     (111111)             (2222)
                                                          (3221)
                                                          (3311)
                                                          (11111111)

Crossrefs

For multiplicities instead of distinct parts we have A360068.
The complement is counted by A360242, ranks A360246.
For median instead of mean we have A360245, complement A360244.
These partitions have ranks A360247.
Cf. A360250 and A360251, ranks A360252 and A360253.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A058398 counts partitions by mean, also A327482.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A360071 counts partitions by number of parts and number of distinct parts.
A360241 counts partitions whose distinct parts have integer mean.
Cf. A051293, A067340, A240219, A316313, A326567/A326568, A326619/A326620, A326621, A349156, A360069.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Mean[#]==Mean[Union[#]]&]],{n,0,30}]
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