A237984 -id:A237984 - cp's OEIS frontend

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A363720 Number of integer partitions of n with different mean, median, and mode.

0, 0, 0, 0, 0, 0, 0, 2, 3, 5, 7, 16, 17, 34, 38, 50, 79, 115, 123, 198, 220, 291, 399, 536, 605, 815, 1036, 1241, 1520, 2059, 2315, 3132, 3708, 4491, 5668, 6587, 7788, 10259, 12299, 14515, 17153, 21558, 24623, 30876, 35540, 41476, 52023, 61931, 70811, 85545

Offset: 0

Gus Wiseman, Jun 21 2023

nonn

If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
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 of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(7) = 2 through a(11) = 16 partitions:
  (421)   (431)   (621)    (532)     (542)
  (3211)  (521)   (3321)   (541)     (632)
          (4211)  (4311)   (631)     (641)
                  (5211)   (721)     (731)
                  (32211)  (5311)    (821)
                           (6211)    (4322)
                           (322111)  (4421)
                                     (5321)
                                     (5411)
                                     (6311)
                                     (7211)
                                     (33221)
                                     (43211)
                                     (52211)
                                     (332111)
                                     (422111)

For equal instead of unequal: A363719, ranks A363727, odd-length A363721.
The case of a unique mode is A363725.
These partitions have ranks A363730.
For factorizations we have A363742, for equal A363741, see A359909, A359910.
Just two statistics:
- (mean) = (median) gives A240219, also A359889, A359895, A359897, A359899.
- (mean) != (median) gives A359894, also A359890, A359896, A359898, A359900.
- (mean) = (mode) gives A363723, see A363724, A363731.
- (median) = (mode) gives A363740.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or negative mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A027193, A237984, A325347, A326567/A326568, A327472, A363726, A363728.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],{Mean[#]}!={Median[#]}!=modes[#]&]],{n,0,30}]

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

Gus Wiseman, Jun 24 2023

nonn

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}.

			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)

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.

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

Gus Wiseman, Jun 30 2023

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

			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)

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.

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

A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 19, 21, 30, 31, 42, 50, 62, 69, 91, 99, 126, 144, 175, 198, 246, 275, 331, 379, 452, 509, 612, 686, 811, 922, 1076, 1219, 1428, 1604, 1863, 2108, 2434, 2739, 3162, 3551, 4075, 4593, 5240, 5885, 6721, 7527, 8556, 9597, 10870

Offset: 0

Reinhard Zumkeller, Jan 20 2010

nonn

A000009(n) <= a(n) <= A000041(n).
Equivalently, the number of partitions of n such that (maximal multiplicity of parts) = (multiplicity of the maximal part), as in the Mathematica program. - Clark Kimberling, Apr 04 2014
Also the number of integer partitions of n whose greatest part is a mode, meaning it appears at least as many times as each of the others. The name "Number of partitions of n such that smaller parts do not occur more frequently than greater parts" seems to describe A100882 = "Number of partitions of n in which the sequence of frequencies of the summands is nonincreasing," which first differs from this at n = 10 due to the partition (3,3,2,1,1). - Gus Wiseman, May 07 2023

			a(5) = #{5, 4+1, 3+2, 2+2+1, 5x1} = 5;
a(6) = #{6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1, 6x1} = 8;
a(7) = #{7, 6+1, 5+2, 4+3, 4+2+1, 3+3+1, 2+2+2+1, 7x1} = 8;
a(8) = #{8, 7+1, 6+2, 5+3, 5+2+1, 4+4, 4+3+1, 3+3+2, 3+3+1+1, 2+2+2+2, 2+2+2+1+1, 8x1} = 12.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions

For median instead of mode we have A053263.
The complement is counted by A240302.
The case where the maximum is the only mode is A362612.
A000041 counts integer partitions, strict A000009.
A362608 counts partitions with a unique mode, complement A362607.
A362611 counts modes in prime factorization.
A362614 counts partitions by number of modes.
Cf. A002865, A008284, A098859, A237984, A275870, A325347, A362610.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *)
Table[Count[f[n], p_ /; m[p] == Count[p, Max[p]]], {n, 0, z}] (* this sequence *)
Table[Count[f[n], p_ /; m[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240302 *)
(* Clark Kimberling, Apr 04 2014 *)
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0],
     If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1,
     If[k == -1, j, If[k == 0, 0, If[j > k, 0, k]]]], {j, 1, n/i}]]];
a[n_] := PartitionsP[n] - b[n, n, -1];
a /@ Range[0, 70] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz in A240302 *)
Table[Length[Select[IntegerPartitions[n],MemberQ[Commonest[#],Max[#]]&]],{n,0,30}] (* Gus Wiseman, May 07 2023 *)

{ my(N=53, x='x+O('x^N));
my(gf=1+sum(i=1,N,sum(j=1,floor(N/i),x^(i*j)*prod(k=1,i-1,(1-x^(k*(j+1)))/(1-x^k)))));
Vec(gf) } \\ John Tyler Rascoe, Mar 09 2024

a(n) = p(n,0,1,1) with p(n,i,j,k) = if k<=n then p(n-k,i,j+1,k) +p(n,max(i,j),1,k+1) else (if j0 then 0 else 1).

a(n) + A240302(n) = A000041(n). - Clark Kimberling, Apr 04 2014.

G.f.: 1 + Sum_{i, j>0} x^(i*j) * Product_{k=1..i-1} ((1 - x^(k*(j+1)))/(1 - x^k)). - John Tyler Rascoe, Mar 09 2024

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

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Author

Gus Wiseman, Jun 30 2023

Keywords

nonn
tabl

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}]

A065795 Number of subsets of {1,2,...,n} that contain the average of their elements.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 26, 42, 72, 124, 218, 390, 706, 1292, 2388, 4436, 8292, 15578, 29376, 55592, 105532, 200858, 383220, 732756, 1403848, 2694404, 5179938, 9973430, 19229826, 37125562, 71762396, 138871260, 269021848, 521666984, 1012520400, 1966957692, 3824240848

Offset: 1

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Author

John W. Layman, Dec 05 2001

Keywords

nonn

Comments

Also the number of subsets of {1,2,...,n} with sum of entries divisible by the largest element (compare A000016). See the Palmer Melbane link for a bijection. - Joel B. Lewis, Nov 13 2014

Examples

a(4)=6, since {1}, {2}, {3}, {4}, {1,2,3} and {2,3,4} contain their averages. From _Gus Wiseman_, Sep 14 2019: (Start) The a(1) = 1 through a(6) = 16 subsets: {1} {1} {1} {1} {1} {1} {2} {2} {2} {2} {2} {3} {3} {3} {3} {1,2,3} {4} {4} {4} {1,2,3} {5} {5} {2,3,4} {1,2,3} {6} {1,3,5} {1,2,3} {2,3,4} {1,3,5} {3,4,5} {2,3,4} {1,2,3,4,5} {2,4,6} {3,4,5} {4,5,6} {1,2,3,6} {1,4,5,6} {1,2,3,4,5} {2,3,4,5,6} (End)

Links

Alois P. Heinz, Table of n, a(n) for n = 1..3333

Palmer Melbane, Combinations under constraint, Art of Problem Solving thread. - _Joel B. Lewis_, Nov 13 2014

Crossrefs

Subsets containing n whose mean is an element are A000016.

The version for integer partitions is A237984.

Subsets not containing their mean are A327471.

Cf. A051293, A063776, A066571, A135342, A324736, A325705, A326083, A326836, A327478, A327481.

Programs

Mathematica

Table[ Sum[a = Select[Divisors[i], OddQ[ # ] &]; Apply[ Plus, 2^(i/a) * EulerPhi[a]]/i, {i, n}]/2, {n, 34}] (* second program *) Table[Length[Select[Subsets[Range[n]],MemberQ[#,Mean[#]]&]],{n,0,10}] (* Gus Wiseman, Sep 14 2019 *)

PARI

a(n) = (1/2)*sum(i=1, n, (1/i)*sumdiv(i, d, if (d%2, 2^(i/d)*eulerphi(d)))); \\ Michel Marcus, Dec 20 2020

Python

from sympy import totient, divisors def A065795(n): return sum((sum(totient(d)<>(~k&k-1).bit_length(),generator=True))<<1)//k for k in range(1,n+1))>>1 # Chai Wah Wu, Feb 22 2023

Formula

a(n) = (1/2)*Sum_{i=1..n} (f(i) - 1) where f(i) = (1/i) * Sum_{d | i and d is odd} 2^(i/d) * phi(d).

a(n) = (n + A051293(n))/2.

a(n) = 2^n - A327471(n). - Gus Wiseman, Sep 14 2019

Extensions

Edited and extended by Robert G. Wilson v, Nov 15 2002

A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 1, 2, 4, 0, 10, 0, 8, 16, 10, 0, 31, 0, 44, 44, 20, 0, 92, 50, 28, 98, 154, 0, 266, 0, 154, 194, 48, 434, 712, 0, 60, 348, 910, 0, 1198, 0, 1120, 2138, 88, 0, 2428, 1300, 1680, 912, 2506, 0, 4808, 4800, 5968, 1372, 140, 0, 14820, 0, 160

Offset: 1

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Author

Gus Wiseman, Mar 29 2023

Keywords

nonn

Comments

Also partitions satisfying (maximum) = 2*(mean).

These are partitions whose diagram has the same size as its complement (see example).

Examples

The a(6) = 2 through a(12) = 10 partitions: (411) . (4211) (621) (5221) . (822) (3111) (321111) (5311) (831) (42211) (6222) (43111) (6321) (6411) (422211) (432111) (441111) (32211111) (33111111) The partition y = (6,4,1,1) has diagram: 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(12).

Crossrefs

For minimum instead of mean we have A118096.

For length instead of mean we have A237753.

For median instead of mean we have A361849, ranks A361856.

This is the equal case of A361851, unequal case A361852.

The strict case is A361854.

These partitions have ranks A361855.

This is the equal case of A361906, unequal case A361907.

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. A111907, A116608, A188814, A237755, A237824, A237984, A240219, A326849, A327482, A349156, A359894.

Programs

Mathematica

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

A363725 Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 3, 8, 8, 17, 19, 28, 39, 59, 68, 106, 123, 165, 220, 301, 361, 477, 605, 745, 929, 1245, 1456, 1932, 2328, 2846, 3590, 4292, 5111, 6665, 8040, 9607, 11532, 14410, 16699, 20894, 24287, 28706, 35745, 42845, 49548, 59963, 70985

Offset: 0

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Author

Gus Wiseman, Jun 22 2023

Keywords

nonn

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 of {a,a,b,b,b,c,d,d,d} are {b,d}.

Examples

The a(7) = 1 through a(13) = 17 partitions: (3211) (4211) (3321) (5311) (4322) (4431) (4432) (4311) (6211) (4421) (5322) (5422) (5211) (322111) (5411) (6411) (5521) (6311) (7311) (6322) (7211) (8211) (6511) (43211) (53211) (7411) (332111) (432111) (8311) (422111) (522111) (9211) (54211) (63211) (333211) (433111) (442111) (532111) (622111) (3322111) (32221111)

Crossrefs

The length-4 case appears to be A325695.

For equal instead of unequal we have A363719, ranks A363727.

Allowing multiple modes gives A363720, ranks A363730.

A000041 counts partitions, strict A000009.

A008284 counts partitions by length (or decreasing mean), strict A008289.

A359893 and A359901 count partitions by median, odd-length A359902.

A362608 counts partitions with a unique mode.

Cf. A237984, A240219, A325347, A326567/A326568, A327472, A359894, A359896, A359900, A363723, A363728.

Programs

Mathematica

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&]; Table[Length[Select[IntegerPartitions[n], Length[modes[#]]==1&&Mean[#]!=Median[#]!=First[modes[#]]&]],{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

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Author

Gus Wiseman, Jan 20 2023

Keywords

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 0..1000

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

A361851 Number of integer partitions of n such that (length) * (maximum) <= 2*n.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 18, 23, 31, 37, 51, 58, 75, 96, 116, 126, 184, 193, 253, 307, 346, 402, 511, 615, 678, 792, 1045, 1088, 1386, 1419, 1826, 2181, 2293, 2779, 3568, 3659, 3984, 4867, 5885, 6407, 7732, 8124, 9400, 11683, 13025, 13269, 16216, 17774, 22016

Offset: 1

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Author

Gus Wiseman, Mar 28 2023

Keywords

nonn

Comments

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

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

Examples

The a(1) = 1 through a(7) = 12 partitions: (1) (2) (3) (4) (5) (6) (7) (11) (21) (22) (32) (33) (43) (111) (31) (41) (42) (52) (211) (221) (51) (61) (1111) (311) (222) (322) (2111) (321) (331) (11111) (411) (421) (2211) (2221) (3111) (3211) (21111) (22111) (111111) (211111) (1111111) The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 <= 2*7, so y is counted under a(7). The partition y = (5,2,1,1) has length 4 and maximum 5, and 4*5 is not <= 2*9, so y is not counted under a(9). The partition y = (3,2,1,1) has diagram: o o o o o . o . . o . . with complement of size 5, and 5 <= 7, so y is counted under a(7).

Crossrefs

For length instead of mean we have A237755.

For minimum instead of mean we have A237824.

For median instead of mean we have A361848.

The equal case for median is A361849, ranks A361856.

The unequal case is A361852, median A361858.

The equal case is A361853, ranks A361855.

Reversing the inequality gives A361906, unequal case A361907.

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, A324521, A324562, A327482, A349156, A360068, A360071, A360241, A361394, A361859.

Programs

Mathematica

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

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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
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A363720 Number of integer partitions of n with different mean, median, and mode.

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

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

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A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

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

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A065795 Number of subsets of {1,2,...,n} that contain the average of their elements.

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A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.

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A363725 Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.

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