A238478 -id:A238478 - cp's OEIS frontend

A237984 Number of partitions of n whose mean is a part.

1, 2, 2, 3, 2, 5, 2, 6, 5, 8, 2, 21, 2, 14, 22, 30, 2, 61, 2, 86, 67, 45, 2, 283, 66, 80, 197, 340, 2, 766, 2, 663, 543, 234, 703, 2532, 2, 388, 1395, 4029, 2, 4688, 2, 4476, 7032, 1005, 2, 17883, 2434, 9713, 7684, 14472, 2, 25348, 17562, 37829, 16786, 3721

Offset: 1

Clark Kimberling, Feb 27 2014

a(n) = 2 if and only if n is a prime.

			a(6) counts these partitions:  6, 33, 321, 222, 111111.
From _Gus Wiseman_, Sep 14 2019: (Start)
The a(1) = 1 through a(10) = 8 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      432        22222
                           321              3221      531        32221
                           111111           4211      111111111  33211
                                            11111111             42211
                                                                 52111
                                                                 1111111111
(End)

Cf. A238478.
The Heinz numbers of these partitions are A327473.
A similar sequence for subsets is A065795.
Dominated by A067538.
The strict case is A240850.
Partitions without their mean are A327472.
Cf. A000016, A316413, A324753, A325705, A327478, A327482.

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Mean[p]]], {n, 40}]

from sympy.utilities.iterables import partitions
def A237984(n): return sum(1 for s,p in partitions(n,size=True) if not n%s and n//s in p) # Chai Wah Wu, Sep 21 2023

a(n) = A000041(n) - A327472(n). - Gus Wiseman, Sep 14 2019

A362608 Number of integer partitions of n having a unique mode.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 7, 11, 16, 21, 29, 43, 54, 78, 102, 131, 175, 233, 295, 389, 490, 623, 794, 1009, 1255, 1579, 1967, 2443, 3016, 3737, 4569, 5627, 6861, 8371, 10171, 12350, 14901, 18025, 21682, 26068, 31225, 37415, 44617, 53258, 63313, 75235, 89173, 105645

Offset: 0

Gus Wiseman, Apr 30 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 of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The partition (3,3,2,1) has greatest multiplicity 2, and a unique part of multiplicity 2 (namely 3), so is counted under a(9).
The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (221)    (33)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (511)
                            (11111)  (3111)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

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

For parts instead of multiplicities we have A000041(n-1), ranks A102750.
For median instead of mode we have A238478, complement A238479.
These partitions have ranks A356862.
The complement is counted by A362607, ranks A362605.
For co-mode complement we have A362609, ranks A362606.
For co-mode we have A362610, ranks A359178.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A053263, A098859, A237984, A304442, A327472, A360071, A360687, A362612.

Table[Length[Select[IntegerPartitions[n],Length[Commonest[#]]==1&]],{n,0,30}]

seq(n) = my(A=O(x*x^n)); Vec(sum(m=1, n, sum(j=1, n\m, x^(j*m)*(1-x^j)/(1 - x^(j*m)), A)*prod(j=1, n\m, (1 - x^(j*m))/(1 - x^j) + A)/prod(j=n\m+1, n, 1 - x^j + A)), -(n+1)) \\ Andrew Howroyd, May 04 2023

G.f.: Sum_{m>=1} (Sum_{j>=1} x^(j*m)*(1 - x^j)/(1 - x^(j*m))) * (Product_{j>=1} (1 - x^(j*m))/(1 - x^j)). - Andrew Howroyd, May 04 2023

A078408 Number of ways to partition 2n+1 into distinct positive integers.

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 27, 38, 54, 76, 104, 142, 192, 256, 340, 448, 585, 760, 982, 1260, 1610, 2048, 2590, 3264, 4097, 5120, 6378, 7917, 9792, 12076, 14848, 18200, 22250, 27130, 32992, 40026, 48446, 58499, 70488, 84756, 101698, 121792, 145578, 173682

Offset: 0

N. J. A. Sloane, Dec 27 2002

a(n) is also the number of partitions of 2n+1 in which all parts are odd, due to Euler's partition theorem. See A000009. - Wolfdieter Lang, Jul 08 2012

			a(3) = 5 because 7 = 1+6 = 2+5 = 3+4 = 1+2+4 (partitions into distinct parts) and 7 = 1+1+5 = 1+3+3 = 1+1+1+1+3 = 1+1+1+1+1+1+1 (partitions into odd parts). [_Wolfdieter Lang_, Jul 08 2012]
G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 18*x^6 + 27*x^7 + 38*x^8 + ...
G.f. = q^25 + 2*q^73 + 3*q^121 + 5*q^169 + 8*q^217 + 12*q^265 + 18*q^313 + ...

G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Reinhard Zumkeller)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035294, A078409, A078410.

Cf. A005408, A000009.

import Data.MemoCombinators (memo2, integral)
a078408 n = a078408_list !! n
a078408_list = f 1 where
   f x = (p' 1 x) : f (x + 2)
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m = if m < k then 0 else p' k (m - k) + p' (k + 2) m
-- Reinhard Zumkeller, Nov 27 2015

G := 1/(1 - x)*add(x^floor(3*n/2)/mul(1 - x^k, k = 1..n), n = 0..50):
S := series(G, x, 76):
seq(coeff(S, x, j), j = 0..75); # Peter Bala, Feb 04 2021

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] / QPochhammer[ x], {x, 0, 2 n + 1}]; (* Michael Somos, Oct 06 2015 *)

{a(n) = my(A); if( n<0, 0, n = 2*n + 1; A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A), n))};

a(n) = t(2*n+1, 0), t as defined in A079211.
Euler transform of period 16 sequence [ 2, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 1, 1, 0, 2, 0, ...]. - Michael Somos, Mar 04 2003
a(n) = A000009(2*n+1). G.f. of A000009: 1/[(1 - x)*(1 - x^3)*(1 - x^5)*...] - Jon Perry, May 27 2004
Expansion of f(x, x^7) / f(-x, -x^2) where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 06 2015
From Peter Bala, Feb 04 2021: (Start)
G.f.: Sum_{n >= 0} x^n/Product_{k = 1..2*n+1} 1 - x^k. Replace q with q^2 and set t = q in Andrews, equation 2.2.5, p. 19, and then take the odd part of the series.
G.f.: 1/(1 - x)*Sum_{n >= 0} x^floor(3*n/2)/Product_{k = 1..n} (1 - x^k). (End)
a(n) = A282893(n+1) + A238478(n+1) = A035294(n+1) - A238479(n+1). - Mathew Englander, May 24 2023
G.f.: Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-1))*(1 + q^(8*n-7))/(1 - q^n). - Peter Bala, Dec 30 2024
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(11/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jul 06 2025

More terms from Reinhard Zumkeller, Dec 28 2002

A362610 Number of integer partitions of n having a unique part of least multiplicity.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 7, 10, 13, 16, 23, 30, 35, 50, 61, 73, 95, 123, 139, 187, 216, 269, 328, 411, 461, 594, 688, 836, 980, 1211, 1357, 1703, 1936, 2330, 2697, 3253, 3649, 4468, 5057, 6005, 6841, 8182, 9149, 10976, 12341, 14508, 16447, 19380, 21611, 25553, 28628

Offset: 0

Gus Wiseman, Apr 30 2023

nonn

Alternatively, these are partitions with a part appearing fewer times than each of the others.

			The partition (3,3,2,2,2,1,1,1) has least multiplicity 2, and only one part of multiplicity 2 (namely 3), so is counted under a(15).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (5111)
                                               (31111)    (22211)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

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

For parts instead of multiplicities we have A002865, ranks A247180.
For median instead of co-mode we have A238478, complement A238479.
These partitions have ranks A359178.
For mode complement of co-mode we have A362607, ranks A362605.
For mode instead of co-mode we have A362608, ranks A356862.
The complement is counted by A362609, ranks A362606.
A000041 counts integer partitions.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A008284, A053263, A098859, A237984, A240219, A304442, A327472, A353863, A353864, A353865, A362612.

Table[Length[Select[IntegerPartitions[n],Count[Length/@Split[#],Min@@Length/@Split[#]]==1&]],{n,0,30}]

seq(n) = my(A=O(x*x^n)); Vec(sum(m=2, n+1, sum(j=1, n, x^(j*(m-1))/(1 + if(j*m<=n, x^(j*m)/(1-x^j) )) + A)*prod(j=1, n\m, 1 + x^(j*m)/(1 - x^j) + A)), -(n+1)) \\ Andrew Howroyd, May 04 2023

G.f.: Sum_{m>=2} (Sum_{j>=1} x^(j*(m-1))/(1 + x^(j*m)/(1 - x^j))) * (Product_{j>=1} (1 + x^(j*m)/(1 - x^j))). - Andrew Howroyd, May 04 2023

A362607 Number of integer partitions of n with more than one mode.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 4, 6, 9, 13, 13, 23, 23, 33, 45, 56, 64, 90, 101, 137, 169, 208, 246, 320, 379, 469, 567, 702, 828, 1035, 1215, 1488, 1772, 2139, 2533, 3076, 3612, 4333, 5117, 6113, 7168, 8557, 10003, 11862, 13899, 16385, 19109, 22525, 26198, 30729, 35736

Offset: 0

Gus Wiseman, Apr 30 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 of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The partition (3,2,2,1,1) has greatest multiplicity 2, and two parts of multiplicity 2 (namely 1 and 2), so is counted under a(9).
The a(3) = 1 through a(9) = 9 partitions:
  (21)  (31)  (32)  (42)    (43)   (53)    (54)
              (41)  (51)    (52)   (62)    (63)
                    (321)   (61)   (71)    (72)
                    (2211)  (421)  (431)   (81)
                                   (521)   (432)
                                   (3311)  (531)
                                           (621)
                                           (32211)
                                           (222111)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 301 terms from John Tyler Rascoe)

For parts instead of multiplicities we have A002865.
For median instead of mode we have A238479, complement A238478.
These partitions have ranks A362605.
The complement is counted by A362608, ranks A356862.
For co-mode we have A362609, ranks A362606.
For co-mode complement we have A362610, ranks A359178.
A000041 counts integer partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A008284, A098859, A237984, A275870, A304442, A327472, A353864, A353865, A360071, A362612.

b:= proc(n, i, m, t) option remember; `if`(n=0, `if`(t=2, 1, 0), `if`(i<1, 0,
      add(b(n-i*j, i-1, max(j, m), `if`(j>m, 1, `if`(j=m, 2, t))), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..51);  # Alois P. Heinz, May 05 2024

Table[Length[Select[IntegerPartitions[n],Length[Commonest[#]]>1&]],{n,0,30}]

G_x(N)={my(x='x+O('x^(N-1)), Ib(k,j) = if(k>j,1,0), A_x(u)=sum(i=1,N-u, sum(j=u+1, N-u, (x^(i*(u+j))*(1-x^u)*(1-x^j))/((1-x^(u*i))*(1-x^(j*i))) * prod(k=1,N-i*(u+j), (1-x^(k*(i+Ib(k,j))))/(1-x^k)))));
concat([0,0,0],Vec(sum(u=1,N, A_x(u))))}
G_x(51) \\ John Tyler Rascoe, Apr 05 2024

G.f.: Sum_{u>0} A(u,x) where A(u,x) = Sum_{i>0} Sum_{j>u} ( x^(i*(u+j))*(1-x^u)*(1-x^j) )/( (1-x^(u*i))*(1-x^(j*i)) ) * Product_{k>0} ( (1-x^(k*(i+[k>j])))/(1-x^k) ) is the g.f. for partitions of this kind with least mode u and [] is the Iverson bracket. - John Tyler Rascoe, Apr 05 2024

A362612 Number of integer partitions of n such that the greatest part is the unique mode.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 6, 6, 7, 9, 10, 12, 15, 16, 19, 23, 26, 32, 37, 41, 48, 58, 65, 75, 88, 101, 115, 135, 151, 176, 200, 228, 261, 300, 336, 385, 439, 498, 561, 641, 717, 818, 921, 1036, 1166, 1321, 1477, 1667, 1867, 2099, 2346, 2640, 2944, 3303, 3684

Offset: 0

Gus Wiseman, May 03 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 of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(1) = 1 through a(10) = 7 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    221    33      331      44        333        55
              1111  11111  222     2221     332       441        442
                           111111  1111111  2222      3321       3331
                                            22211     22221      22222
                                            11111111  111111111  222211
                                                                 1111111111

John Tyler Rascoe, Table of n, a(n) for n = 0..500

For median instead of mode we have A053263, complement A237821.
These partitions have ranks A362616.
A000041 counts integer partitions.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362607 counts partitions with more than one mode, ranks A362605.
A362608 counts partitions with a unique mode, ranks A356862.
A362611 counts modes in prime factorization.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A070003, A098859, A102750, A237984, A238478, A238479, A327472, A362609, A362610.

Table[Length[Select[IntegerPartitions[n],Commonest[#]=={Max[#]}&]],{n,0,30}]

A_x(N)={my(x='x+O('x^N), g=sum(i=1, N, sum(j=1, N/i, x^(i*j)*prod(k=1,i-1,(1-x^(j*k))/(1-x^k))))); concat([0],Vec(g))}
A_x(60) \\ John Tyler Rascoe, Apr 03 2024

G.f.: Sum_{i, j>0} x^(i*j) * Product_{k=1,i-1} ((1-x^(j*k))/(1-x^k)). - John Tyler Rascoe, Apr 03 2024

A362609 Number of integer partitions of n with more than one part of least multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 19, 26, 42, 51, 74, 103, 136, 174, 246, 303, 411, 523, 674, 844, 1114, 1364, 1748, 2174, 2738, 3354, 4247, 5139, 6413, 7813, 9613, 11630, 14328, 17169, 20958, 25180, 30497, 36401, 44025, 52285, 62834, 74626, 89111, 105374, 125662

Offset: 0

Gus Wiseman, Apr 30 2023

nonn

These are partitions where no part appears fewer times than all of the others.

			The partition (4,2,2,1) has least multiplicity 1, and two parts of multiplicity 1 (namely 1 and 4), so is counted under a(9).
The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)
              (41)  (51)    (52)    (62)     (63)
                    (321)   (61)    (71)     (72)
                    (2211)  (421)   (431)    (81)
                            (3211)  (521)    (432)
                                    (3221)   (531)
                                    (3311)   (621)
                                    (4211)   (3321)
                                    (32111)  (4221)
                                             (4311)
                                             (5211)
                                             (42111)
                                             (222111)
                                             (321111)

For parts instead of multiplicities we have A117989, ranks A283050.
For median instead of co-mode we have A238479, complement A238478.
These partitions have ranks A362606.
For mode instead of co-mode we have A362607, ranks A362605.
For mode complement instead of co-mode we have A362608, ranks A356862.
The complement is counted by A362610, ranks A359178.
A000041 counts integer partitions.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A053263, A098859, A304442, A353864, A360071, A362612.

Table[Length[Select[IntegerPartitions[n],Count[Length/@Split[#],Min@@Length/@Split[#]]>1&]],{n,0,30}]

A237824 Number of partitions of n such that 2*(least part) >= greatest part.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 10, 11, 13, 14, 19, 18, 23, 25, 29, 30, 38, 37, 46, 48, 54, 57, 70, 69, 80, 85, 97, 100, 118, 118, 137, 144, 159, 168, 193, 195, 220, 233, 259, 268, 303, 311, 348, 367, 399, 419, 469, 483, 532, 560, 610, 639, 704, 732, 801, 841, 908, 954

Offset: 1

Clark Kimberling, Feb 16 2014

nonn

By conjugation, also the number of integer partitions of n whose greatest part appears at a middle position, namely at k/2, (k+1)/2, or (k+2)/2 where k is the number of parts. These partitions have ranks A362622. - Gus Wiseman, May 14 2023

			a(6) = 7 counts these partitions:  6, 42, 33, 222, 2211, 21111, 111111.
From _Gus Wiseman_, May 14 2023: (Start)
The a(1) = 1 through a(8) = 10 partitions such that 2*(least part) >= greatest part:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (42)      (322)      (53)
                    (1111)  (2111)   (222)     (2221)     (332)
                            (11111)  (2211)    (22111)    (422)
                                     (21111)   (211111)   (2222)
                                     (111111)  (1111111)  (22211)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part appears at a middle position:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (2211)    (2221)     (332)
                                     (111111)  (1111111)  (2222)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..300 from John Tyler Rascoe)

Cf. A237821, A118096, A053263.
The complement is counted by A237820, ranks A362982.
For modes instead of middles we have A362619, counted by A171979.
These partitions have ranks A362981.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
Cf. A002865, A008284, A237984, A238478, A238479, A327472, A359893, A362612, A362622, A384426.

z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] == Max[p]], {n, z}] (* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* this sequence *)
(* or *)
nmax = 100; Rest[CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j,k,2*k}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 13 2025 *)
(* or *)
nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(2*k - 1))*(1 - x^(2*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax+1)]; s += x^k/(1 - x^k)/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 14 2025 *)

N=60; x='x+O('x^N);
gf = sum(m=1, N, (x^m)/(1-x^m)) + sum(i=1, N, sum(j=1, i, x^((2*i)+j)/prod(k=0, j, 1 - x^(k+i))));
Vec(gf) \\ John Tyler Rascoe, Mar 07 2024

G.f.: Sum_{m>0} x^m/(1-x^m) + Sum_{i>0} Sum_{j=1..i} x^((2*i)+j) / Product_{k=0..j} (1 - x^(k+i)). - John Tyler Rascoe, Mar 07 2024
G.f.: Sum_{k>=1} x^k / Product_{j=k..2*k} (1 - x^j). - Vaclav Kotesovec, Jun 13 2025
a(n) ~ phi^(3/2) * exp(Pi*sqrt(2*n/15)) / (5^(1/4) * sqrt(2*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 14 2025

A238479 Number of partitions of n whose median is not a part.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 5, 8, 10, 13, 18, 23, 30, 40, 50, 64, 83, 104, 131, 166, 206, 256, 320, 394, 485, 598, 730, 891, 1088, 1318, 1596, 1932, 2326, 2797, 3360, 4020, 4804, 5735, 6824, 8108, 9624, 11392, 13468, 15904, 18737, 22048, 25914, 30400, 35619, 41686

Offset: 1

Clark Kimberling, Feb 27 2014

Also, the number of partitions p of n such that (1/2)*max(p) is a part of p.

Also the number of even-length integer partitions of n with distinct middle parts. For example, the partition (4,3,2,1) has middle parts {2,3} so is counted under a(10), but (3,2,2,1) has middle parts {2,2} so is not counted under a(8). - Gus Wiseman, May 13 2023

			a(6) counts these partitions:  51, 42, 2211 which all have an even number of parts, and their medians 3, 3 and 1.5 are not present. Note that the partitions 33 and 3111, although having an even number of parts, are not included in the count of a(6), but instead in that of A238478(6), as their medians, 3 for the former and 1 for the latter, are present in those partitions.

A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.

The complement is A238478, ranks A362618.
For mean instead of median we have A327472, complement A237984.
These partitions have ranks A362617.
A000041 counts integer partitions, even-length A027187.
A325347 counts partitions with integer median, complement A307683.
A359893/A359901/A359902 count partitions by median.
A359908 ranks partitions with integer median, complement A359912.
Cf. A002865, A008284, A053263, A238480, A238481, A240219, A359907, A360686, A360687, A360952.

Table[Count[IntegerPartitions[n], p_ /; !MemberQ[p, Median[p]]], {n, 40}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[p]/2]], {n, 50}]

my(q='q+O('q^50)); concat([0,0], Vec(sum(n=1,17,q^(3*n)/prod(k=1,2*n,1-q^k)))) \\ David Radcliffe, Jun 25 2025

from sympy.utilities.iterables import partitions
def A238479(n): return sum(1 for p in partitions(n) if (m:=max(p,default=0))&1^1 and m>>1 in p) # Chai Wah Wu, Sep 21 2023

a(n) = A000041(n) - A238478(n).
For all n, A027187(n) >= a(n). [Because when a partition of n has an odd number of parts, then it is not counted by this sequence (cf. A238478) and also some of the partitions with an even number of parts might be excluded here. Cf. Examples.] - Antti Karttunen, Feb 27 2014
From Jeremy Lovejoy, Sep 29 2022: (Start)
G.f.: Sum_{n>=1} q^(3*n)/Product_{k=1..2*n} (1-q^k).
a(n) ~ Pi/(2^(17/4)*3^(3/4)*n^(5/4))*exp(Pi*sqrt(2*n/3)). Proved by Blecher and Knopfmacher. (End)
a(n) = A087897(2*n) = A035294(n) - A078408(n-1). - Mathew Englander, May 20 2023

A363719 Number of integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 58, 15, 18, 37, 60, 2, 123, 2, 98, 79, 35, 103, 332, 2, 49, 166, 451, 2, 515, 2, 473, 738, 92, 2, 1561, 277, 839, 631, 1234, 2, 2043, 1560, 2867, 1156, 225, 2, 9020

Offset: 1

Gus Wiseman, Jun 19 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).
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}.
Without loss of generality, we may assume there is a unique middle-part (A238478).
Includes all constant partitions.

			The a(n) partitions for n = 1, 2, 4, 6, 8, 12, 14, 16 (A..G = 10..16):
  1  2   4     6       8         C             E               G
     11  22    33      44        66            77              88
         1111  222     2222      444           2222222         4444
               111111  3221      3333          3222221         5443
                       11111111  4332          3322211         6442
                                 5331          4222211         7441
                                 222222        11111111111111  22222222
                                 322221                        32222221
                                 422211                        33222211
                                 111111111111                  42222211
                                                               52222111
                                                               1^16

For unequal instead of equal: A363720, ranks A363730, unique mode A363725.
The odd-length case is A363721.
These partitions have ranks A363727, nonprime A363722.
The case of non-constant partitions is A363728, ranks A363729.
The version for factorizations is 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, A363742.

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

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A237984 Number of partitions of n whose mean is a part.

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A362608 Number of integer partitions of n having a unique mode.

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A078408 Number of ways to partition 2n+1 into distinct positive integers.

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A362610 Number of integer partitions of n having a unique part of least multiplicity.

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A362607 Number of integer partitions of n with more than one mode.

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A362612 Number of integer partitions of n such that the greatest part is the unique mode.

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A362609 Number of integer partitions of n with more than one part of least multiplicity.

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