A047993 -id:A047993 - cp's OEIS frontend

A064174 Number of partitions of n with nonnegative rank.

1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 31, 42, 56, 73, 96, 125, 161, 207, 265, 336, 426, 536, 672, 840, 1046, 1296, 1603, 1975, 2425, 2970, 3628, 4417, 5367, 6503, 7861, 9482, 11412, 13702, 16423, 19642, 23447, 27938, 33231, 39453, 46767, 55342, 65386, 77135

Offset: 1

Vladeta Jovovic, Sep 20 2001

nonn

The rank of a partition is the largest summand minus the number of summands.
This sequence (up to proof) equals "partitions of 2n with even number of parts, ending in 1, with max descent of 1, where the number of odd parts in odd places equals the number of odd parts in even places. (See link and 2nd Mathematica line.) - Wouter Meeussen, Mar 29 2013
Number of partitions p of n such that max(max(p), number of parts of p) is a part of p. - Clark Kimberling, Feb 28 2014
From Gus Wiseman, Mar 09 2019: (Start)
Also the number of integer partitions of n with maximum part greater than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324521. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)
            (21)  (22)  (32)   (33)   (43)    (44)
                  (31)  (41)   (42)   (52)    (53)
                        (311)  (51)   (61)    (62)
                               (321)  (322)   (71)
                               (411)  (331)   (332)
                                      (421)   (422)
                                      (511)   (431)
                                      (4111)  (521)
                                              (611)
                                              (4211)
                                              (5111)
Also the number of integer partitions of n with maximum part less than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324562. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (11)  (21)   (22)    (221)    (222)     (322)      (332)
             (111)  (211)   (311)    (321)     (331)      (2222)
                    (1111)  (2111)   (2211)    (2221)     (3221)
                            (11111)  (3111)    (3211)     (3311)
                                     (21111)   (4111)     (4211)
                                     (111111)  (22111)    (22211)
                                               (31111)    (32111)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

			a(20) = p(19) - p(15) + p(8) = 490 - 176 + 22 = 336.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Cristina Ballantine and Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.
Rekha Biswal, bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n, Mathoverflow.
Mircea Merca, Rank partition functions and truncated theta identities, arXiv:2006.07705 [math.CO], 2020.

Cf. A063995, A064173 (complement).
Row sums of triangle A105806.
Cf. A003114, A006141, A039900, A047993, A090858, A106529, A133121.
Cf. A324516, A324518, A324520, A324521, A324522, A324560, A324562, A324572.

f:= n -> add((-1)^(k+1)*combinat:-numbpart(n-(3*k^2-k)/2),k=1..floor((1+sqrt(24*n+1))/6)):
map(f, [$1..100]); # Robert Israel, Aug 03 2015

Table[Count[IntegerPartitions[n], q_ /; First[q] >= Length[q]], {n, 16}]
(* also *)
Table[Count[IntegerPartitions[2n],q_/;Last[q]===1 && Max[q-PadRight[Rest[q],Length[q]]]<=1 && Count[First/@Partition[q,2],?OddQ]==Count[Last/@Partition[q,2],?OddQ]],{n,16}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p], Length[p]]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)

{a(n) = my(A=1); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}
for(n=1,60,print1(a(n),", ")) \\ Paul D. Hanna, Aug 03 2015

my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2))) \\ Seiichi Manyama, May 21 2023

a(n) = (A000041(n) + A047993(n))/2.
a(n) = p(n-1) - p(n-5) + p(n-12) - ... -(-1)^k*p(n-(3*k^2-k)/2) + ..., where p() is A000041(). - Vladeta Jovovic, Aug 04 2004
G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(n+k-1))/(1 - x^k). - Paul D. Hanna, Aug 03 2015
A064173(n) + a(n) = A000041(n). - R. J. Mathar, Feb 22 2023
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2). - Seiichi Manyama, May 21 2023

Mathematica programs modified by Clark Kimberling, Feb 12 2014

A340653 Number of balanced factorizations of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 1, 0, 0, 1, 2, 1, 3, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 1, 3, 1, 2, 2, 0, 1, 2, 0, 2, 0, 2, 1, 1, 0, 1, 0, 0, 1, 2, 1, 0, 2, 1, 0, 3, 1, 2, 0, 3, 1, 3, 1, 0, 2, 2, 0, 3, 1, 2, 1, 0, 1, 2, 0, 0, 0, 1, 1, 2, 0, 2, 0, 0, 0, 3, 1, 2, 2, 2, 1, 3, 1, 1, 3, 0, 1, 3, 1, 3, 0, 2, 1, 3, 0, 2, 2, 0, 0, 4

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

A factorization into factors > 1 is balanced if it is empty or its length is equal to its maximum Omega (A001222).

			The balanced factorizations for n = 120, 144, 192, 288, 432, 768:
  3*5*8    2*8*9    3*8*8      4*8*9      6*8*9      8*8*12
  2*2*30   3*6*8    4*6*8      6*6*8      2*8*27     2*2*8*24
  2*3*20   2*4*18   2*8*12     2*8*18     3*8*18     2*3*8*16
  2*5*12   2*6*12   4*4*12     3*8*12     4*4*27     2*4*4*24
           3*4*12   2*2*2*24   4*4*18     4*6*18     2*4*6*16
                    2*2*3*16   4*6*12     4*9*12     3*4*4*16
                               2*12*12    6*6*12     2*2*12*16
                               2*2*2*36   2*12*18    2*2*2*2*48
                               2*2*3*24   3*12*12    2*2*2*3*32
                               2*3*3*16   2*2*2*54
                                          2*2*3*36
                                          2*3*3*24
                                          3*3*3*16

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Positions of zeros are A001358.
Positions of nonzero terms are A100959.
The co-balanced version is A340596.
Taking maximum factor instead of maximum Omega gives A340599.
The cross-balanced version is A340654.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A320655 counts factorizations into semiprimes.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 have an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
- A340656 have no twice-balanced factorizations.
- A340657 have a twice-balanced factorization.
Cf. A003963, A117409, A303975, A320656, A339846, A339890, A340608.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],#=={}||Length[#]==Max[PrimeOmega/@#]&]],{n,100}]

A340653(n, m=n, mbo=0, e=0) = if(1==n, mbo==e, sumdiv(n, d, if((d>1)&&(d<=m), A340653(n/d, d, max(mbo,bigomega(d)), 1+e)))); \\ Antti Karttunen, Oct 22 2023

Data section extended up to a(120) by Antti Karttunen, Oct 22 2023

A326836 Heinz numbers of integer partitions whose maximum part divides their sum.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 70, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 135, 137, 139, 144, 149, 150, 151

Offset: 1

Gus Wiseman, Jul 26 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose maximum prime index divides their sum of prime indices.

The enumeration of these partitions by sum is given by A067538.

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}
   31: {11}
   32: {1,1,1,1,1}

Cf. A018818, A047993, A056239, A061395, A067538, A112798, A316413.

Cf. A326837, A326838, A326839/A326840, A326841, A326843, A326850.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],Divisible[Total[primeMS[#]],Max[primeMS[#]]]&]

A384175 Number of subsets of {1..n} with all distinct lengths of maximal runs (increasing by 1).

Original entry on oeis.org

1, 2, 4, 7, 13, 24, 44, 77, 135, 236, 412, 713, 1215, 2048, 3434, 5739, 9559, 15850, 26086, 42605, 69133, 111634, 179602, 288069, 460553, 733370, 1162356, 1833371, 2878621, 4501856, 7016844, 10905449, 16904399, 26132460, 40279108, 61885621, 94766071, 144637928

Offset: 0

Gus Wiseman, Jun 16 2025

nonn

			The subset {2,3,5,6,7,9} has maximal runs ((2,3),(5,6,7),(9)), with lengths (2,3,1), so is counted under a(9).
The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {2,3}    {1,2}
                  {1,2,3}  {2,3}
                           {3,4}
                           {1,2,3}
                           {1,2,4}
                           {1,3,4}
                           {2,3,4}
                           {1,2,3,4}

Christian Sievers, Table of n, a(n) for n = 0..1000

For equal instead of distinct lengths we have A243815.
These subsets are ranked by A328592.
The complement is counted by A384176.
For anti-runs instead of runs we have A384177, ranks A384879.
For partitions instead of subsets we have A384884, A384178, A384886, A384880.
For permutations instead of subsets we have A384891, equal instead of distinct A384892.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000009, A010027, A044813, A047993, A242882, A325325, A329739, A351202, A383013, A384889, A384890.

Table[Length[Select[Subsets[Range[n]],UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

lista(n)={my(o=(1-x^(n+1))/(1-x)*O(y^(n+2)),p=prod(i=1,n,1+o+x*y^(i+1)/(1-y),1/(1-y)));p=subst(serlaplace(p),x,1);Vec(p-1)} \\ Christian Sievers, Jun 18 2025

a(21) and beyond from Christian Sievers, Jun 18 2025

A349156 Number of integer partitions of n whose mean is not an integer.

Original entry on oeis.org

1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752

Offset: 0

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

By conjugation, also the number of integer partitions of n with greatest part not dividing n.

			The a(3) = 1 through a(8) = 11 partitions:
  (21)  (211)  (32)    (2211)   (43)      (332)
               (41)    (3111)   (52)      (422)
               (221)   (21111)  (61)      (431)
               (311)            (322)     (521)
               (2111)           (331)     (611)
                                (421)     (22211)
                                (511)     (32111)
                                (2221)    (41111)
                                (3211)    (221111)
                                (4111)    (311111)
                                (22111)   (2111111)
                                (31111)
                                (211111)

Below, "!" means either enumerative or set theoretical complement.
The version for nonempty subsets is !A051293.
The complement is counted by A067538, ranked by A316413.
The geometric version is !A067539, strict !A326625, ranked by !A326623.
The strict case is !A102627.
The version for prime factors is A175352, complement A078175.
The version for distinct prime factors is A176587, complement A078174.
The ordered version (compositions) is !A271654, ranked by !A096199.
The multiplicative version (factorizations) is !A326622, geometric !A326028.
The conjugate is ranked by !A326836.
The conjugate strict version is !A326850.
These partitions are ranked by A348551.
A000041 counts integer partitions.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A236634 counts unbalanced partitions, complement of A047993.
A327472 counts partitions not containing their mean, complement of A237984.
Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.

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

a(n > 0) = A000041(n) - A067538(n).

A160786 The number of odd partitions of consecutive odd integers.

Original entry on oeis.org

1, 2, 4, 8, 16, 29, 52, 90, 151, 248, 400, 632, 985, 1512, 2291, 3431, 5084, 7456, 10836, 15613, 22316, 31659, 44601, 62416, 86809, 120025, 165028, 225710, 307161, 416006, 560864, 752877, 1006426, 1340012, 1777365, 2348821, 3093095, 4059416, 5310255, 6924691

Offset: 0

Utpal Sarkar (doetoe(AT)gmail.com), May 26 2009

nonn

It seems that these are partitions of odd length and sum, ranked by A340931. The parts do not have to be odd. - Gus Wiseman, Apr 06 2021

			From _Gus Wiseman_, Apr 06 2021: (Start)
The a(0) = 1 through a(4) = 16 partitions:
  (1)  (3)    (5)      (7)        (9)
       (111)  (221)    (322)      (333)
              (311)    (331)      (432)
              (11111)  (421)      (441)
                       (511)      (522)
                       (22111)    (531)
                       (31111)    (621)
                       (1111111)  (711)
                                  (22221)
                                  (32211)
                                  (33111)
                                  (42111)
                                  (51111)
                                  (2211111)
                                  (3111111)
                                  (111111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..5000
V. I. Arnold, On teaching mathematics

Partitions with all odd parts are counted by A000009 and ranked by A066208.
This is a bisection of A027193 (odd-length partitions), which is ranked by A026424.
The case of all odd parts is counted by A078408 and ranked by A300272.
The even version is A236913, ranked by A340784.
A multiplicative version is A340102.
These partitions are ranked by A340931.
A047993 counts balanced partitions, ranked by A106529.
A058695 counts partitions of odd numbers, ranked by A300063.
A072233 counts partitions by sum and length.
A236914 counts partition of type OO, ranked by A341448.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Cf. A000700, A168659, A200750, A340604, A340607, A340854, A340855.

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3],
      `if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p->
      `if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]],
          [p[2], p[1], p[4], p[3]]))(b(n-i, i)))))
    end:
a:= n-> b(2*n+1$2)[2]:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 16 2014

b[n_, i_] := b[n, i] = If[n==0, {1, 0, 0, 0}, If[i<1, {0, 0, 0, 0}, b[n, i-1] + If[i>n, {0, 0, 0, 0}, Function[{p}, If[Mod[i, 2]==0, p[[{3, 4, 1, 2}]], p[[{2, 1, 4, 3}]]]][b[n-i, i]]]]]; a[n_] := b[2*n+1, 2*n+1][[2]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 01 2015, after Alois P. Heinz *)
(* Slow but easy to read *)
a[n_] := Length@IntegerPartitions[2 n + 1, {1, 2 n + 1, 2}]
a /@ Range[0, 25]
(* Leo C. Stein, Nov 11 2020 *)
(* Faster, don't build the partitions themselves *)
(* Number of partitions of n into exactly k parts *)
P[0, 0] = 1;
P[n_, k_] := 0 /; ((k <= 0) || (n <= 0))
P[n_, k_] := P[n, k] = P[n - k, k] + P[n - 1, k - 1]
a[n_] := Sum[P[2 n + 1, k], {k, 1, 2 n + 1, 2}]
a /@ Range[0, 40]
(* Leo C. Stein, Nov 11 2020 *)

# Could be memoized for speedup
def numoddpart(n, m=1):
    """The number of partitions of n into an odd number of parts of size at least m"""
    if n < m:
        return 0
    elif n == m:
        return 1
    else:
        # 1 (namely n = n) and all partitions of the form
        # k + even partitions that start with >= k
        return 1 + sum([numevenpart(n - k,  k) for k in range(m, n//3 + 1)])
def numevenpart(n, m=1):
    """The number of partitions of n into an even number of parts of size at least m"""
    if n < 2*m:
        return 0
    elif n == 2*m:
        return 1
    else:
        return sum([numoddpart(n - k,  k) for k in range(m,  n//2 + 1)])
[numoddpart(n) for n in range(1, 70, 2)]

# dict to memoize
ps = {(0,0): 1}
def p(n, k):
    """Number of partitions of n into exactly k parts"""
    if (n,k) in ps: return ps[(n,k)]
    if (n<=0) or (k<=0): return 0
    ps[(n,k)] = p(n-k,k) + p(n-1,k-1)
    return ps[(n,k)]
def a(n): return sum([p(2*n+1, k) for k in range(1,2*n+3,2)])
[a(n) for n in range(0,41)]
# Leo C. Stein, Nov 11 2020

a(n) = A027193(2n+1).

A168659 Number of partitions of n such that the number of parts is divisible by the greatest part. Also number of partitions of n such that the greatest part is divisible by the number of parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 6, 8, 9, 14, 16, 22, 25, 33, 39, 51, 60, 79, 92, 116, 137, 174, 204, 254, 300, 368, 435, 530, 625, 760, 896, 1076, 1267, 1518, 1780, 2121, 2484, 2946, 3444, 4070, 4749, 5594, 6514, 7637, 8879, 10384, 12043, 14040, 16255

Offset: 1

Vladeta Jovovic, Dec 02 2009

nonn

			a(5)=3 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,1,3] the number of parts is divisible by the greatest part; not true for the partitions [1,2,2],[2,3], [1,4], and [5]. - _Emeric Deutsch_, Dec 04 2009
From _Gus Wiseman_, Feb 08 2021: (Start)
The a(1) = 1 through a(10) = 9 partitions of the first type:
  1  11  21   22    311    321     322      332       333        4222
         111  1111  2111   2211    331      2222      4221       4321
                    11111  111111  2221     4211      4311       4411
                                   4111     221111    51111      52111
                                   211111   311111    222111     222211
                                   1111111  11111111  321111     322111
                                                      21111111   331111
                                                      111111111  22111111
                                                                 1111111111
The a(1) = 1 through a(11) = 14 partitions of the second type (A=10, B=11):
  1   2   3    4    5     6     7      8      9       A       B
          21   22   41    42    43     44     63      64      65
                    311   321   61     62     81      82      83
                                322    332    333     622     A1
                                331    611    621     631     632
                                4111   4211   4221    4222    641
                                              4311    4321    911
                                              51111   4411    4322
                                                      52111   4331
                                                              4421
                                                              8111
                                                              52211
                                                              53111
                                                              611111
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..5000 (terms 1..301 from Vladeta Jovovic corrected by N. J. A. Sloane, Oct 05 2010, terms 302..1000 from Seiichi Manyama)

Cf. A168656, A168657, A079501, A168655.
Note: A-numbers of Heinz-number sequences are in parentheses below.
The case of equality is A047993 (A106529).
The Heinz numbers of these partitions are A340609/A340610.
If all parts (not just the greatest) are divisors we get A340693 (A340606).
The strict case in the second interpretation is A340828 (A340856).
A006141 = partitions whose length equals their minimum (A324522).
A067538 = partitions whose length/max divides their sum (A316413/A326836).
A200750 = partitions with length coprime to maximum (A340608).
Cf. A003114, A039900, A064173, A064174, A143773, A326837, A340653, A340830, A340851, A340853.
Row sums of A350879.

a := proc (n) local pn, ct, j: with(combinat): pn := partition(n): ct := 0: for j to numbpart(n) do if `mod`(nops(pn[j]), max(seq(pn[j][i], i = 1 .. nops(pn[j])))) = 0 then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 50); # Emeric Deutsch, Dec 04 2009

Table[Length[Select[IntegerPartitions[n],Divisible[Length[#],Max[#]]&]],{n,30}] (* Gus Wiseman, Feb 08 2021 *)
nmax = 100; s = 0; Do[s += Normal[Series[Sum[x^((m+1)*k - 1) * Product[(1 - x^(m*k + j - 1))/(1 - x^j), {j, 1, k-1}], {k, 1, (1 + nmax)/(1 + m) + 1}], {x, 0, nmax}]], {m, 1, nmax}]; Rest[CoefficientList[s, x]] (* Vaclav Kotesovec, Oct 18 2024 *)

G.f.: Sum_{i>=1} Sum_{j>=1} x^((i+1)*j-1) * Product_{k=1..j-1} (1-x^(i*j+k-1))/(1-x^k). - Seiichi Manyama, Jan 24 2022

a(n) ~ c * exp(Pi*sqrt(2*n/3)) / n^(3/2), where c = 0.04628003... - Vaclav Kotesovec, Nov 16 2024

Extended by Emeric Deutsch, Dec 04 2009

A064173 Number of partitions of n with positive rank.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 35, 45, 62, 80, 106, 136, 178, 225, 291, 366, 466, 583, 735, 912, 1140, 1407, 1743, 2140, 2634, 3214, 3932, 4776, 5807, 7022, 8495, 10225, 12313, 14762, 17696, 21136, 25236, 30030, 35722, 42367, 50216, 59368, 70138, 82665

Offset: 1

Vladeta Jovovic, Sep 19 2001

nonn

The rank of a partition is the largest summand minus the number of summands.
Also number of partitions of n with negative rank. - Omar E. Pol, Mar 05 2012
Column 1 of A208478. - Omar E. Pol, Mar 11 2012
Number of partitions p of n such that max(max(p), number of parts of p) is not a part of p. - Clark Kimberling, Feb 28 2014
The sequence enumerates the semigroup of partitions of positive rank for each number n. The semigroup is a subsemigroup of the monoid of partitions of nonnegative rank under the binary operation "*": Let A be the positive rank partition (a1,...,ak) where ak > k, and let B=(b1,...bj) with bj > j. Then let A*B be the partition (a1b1,...,a1bj,...,akb1,...,akbj), which has akbj > kj, thus having positive rank. For example, the partition (2,3,4) of 9 has rank 1, and its product with itself is (4,6,6,8,8,9,12,12,16) of 81, which has rank 7. A similar situation holds for partitions of negative rank--they are a subsemigroup of the monoid of nonpositive rank partitions. - Richard Locke Peterson, Jul 15 2018

			a(20) = p(18) - p(13) + p(5) = 385 - 101 + 7 = 291.
From _Gus Wiseman_, Feb 09 2021: (Start)
The a(2) = 1 through a(9) = 13 partitions of positive rank:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (31)  (32)  (33)   (43)   (44)    (54)
                  (41)  (42)   (52)   (53)    (63)
                        (51)   (61)   (62)    (72)
                        (411)  (421)  (71)    (81)
                               (511)  (422)   (432)
                                      (431)   (441)
                                      (521)   (522)
                                      (611)   (531)
                                      (5111)  (621)
                                              (711)
                                              (5211)
                                              (6111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000
F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.
FindStat, St000145: The Dyson rank of a partition
Mircea Merca, Rank partition functions and truncated theta identities, arXiv:2006.07705 [math.CO], 2020.

Note: A-numbers of ranking sequences are in parentheses below.
The negative-rank version is also A064173 (A340788).
The case of odd positive rank is A101707 (A340604).
The case of even positive rank is A101708 (A340605).
These partitions are ranked by (A340787).
A063995/A105806 count partitions by rank.
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is a multiple of the greatest part.
A200750 counts partitions whose length and greatest part are coprime.
- Rank -
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
- Balance -
A047993 counts balanced partitions (A106529).
A340599 counts alt-balanced factorizations.
A340653 counts balanced factorizations.
Cf. A003114, A006141, A039900, A096401, A117193, A117409, A143773, A324516, A324518, A324520.

A064173 := proc(n)
    a := 0 ;
    for p in combinat[partition](n) do
        r := max(op(p))-nops(p) ;
        if r > 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A064173(n),n=0..40) ;# Emeric Deutsch, Dec 11 2004

Table[Count[IntegerPartitions[n], q_ /; First[q] > Length[q]], {n, 24}] (* Clark Kimberling, Feb 12 2014 *)
Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Max[Max[p], Length[p]]]], {n, 20}] (* Clark Kimberling, Feb 28 2014 *)
P = PartitionsP;
a[n_] := (P[n] - Sum[-(-1)^k (P[n - (3k^2 - k)/2] - P[n - (3k^2 + k)/2]), {k, 1, Floor[(1 + Sqrt[1 + 24n])/6]}])/2;
a /@ Range[48] (* Jean-François Alcover, Jan 11 2020, after Wouter Meeussen in A047993 *)

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(k+j-2))/(1-x^j))))) \\ Seiichi Manyama, Jan 25 2022

a(n) = (A000041(n) - A047993(n))/2.
a(n) = p(n-2) - p(n-7) + p(n-15) - ... - (-1)^k*p(n-(3*k^2+k)/2) + ..., where p() is A000041(). - Vladeta Jovovic, Aug 04 2004
G.f.: Product_{k>=1} (1/(1-q^k)) * Sum_{k>=1} ( (-1)^k * (-q^(3*k^2/2+k/2))) (conjectured). - Thomas Baruchel, May 12 2018
G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(k+j-2))/(1-x^j). - Seiichi Manyama, Jan 25 2022
a(n)+A064174(n) = A000041(n). - R. J. Mathar, Feb 22 2023

A268193 Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of partitions of n which have k distinct parts i such that i+1 is also a part.

Original entry on oeis.org

1, 2, 2, 1, 4, 1, 4, 3, 8, 2, 1, 8, 6, 1, 13, 7, 2, 15, 11, 4, 22, 15, 4, 1, 24, 24, 7, 1, 37, 26, 12, 2, 40, 42, 16, 3, 57, 50, 22, 6, 64, 72, 33, 6, 1, 89, 84, 46, 11, 1, 98, 122, 60, 15, 2, 135, 141, 82, 24, 3, 149, 198, 106, 32, 5, 199, 231, 144, 45, 8, 224, 309, 187, 61, 10, 1

Offset: 1

Emeric Deutsch, Feb 13 2016

T(n,k) = number of partitions of n having k singleton parts other than the largest part. Example: T(5,1) = 3 because we have [4,1'], [3,2'], [2,2,1'] (the counted singletons are marked). These partitions are connected by conjugation to those in the definition.
From Gus Wiseman, Jul 10 2025: (Start)
Also the number of integer partitions of n with k maximal subsequences of consecutive parts not decreasing by 1 (anti-runs). For example, row n = 8 counts partitions with the following anti-runs:
  ((8))                ((3,3),(2))          ((3),(2,2),(1))
  ((4,4))              ((4),(3,1))          ((3),(2),(1,1,1))
  ((5,3))              ((5,2),(1))
  ((6,2))              ((4,2),(1,1))
  ((7,1))              ((2,2,2),(1,1))
  ((4,2,2))            ((2,2),(1,1,1,1))
  ((6,1,1))            ((2),(1,1,1,1,1,1))
  ((2,2,2,2))
  ((3,3,1,1))
  ((5,1,1,1))
  ((4,1,1,1,1))
  ((3,1,1,1,1,1))
  ((1,1,1,1,1,1,1,1))
(End)

			T(5,1) = 3 because we have [3,2], [2,2,1], and [2,1,1,1].
T(9,2) = 4 because we have [3,2',1,1,1,1'], [3,2,2',1,1'], [3,3,2',1'], and [4,3',2'] (the i's are marked).
Triangle starts:
  1;
  2;
  2,1;
  4,1;
  4,3;
  8,2,1;
  8,6,1;
From _Gus Wiseman_, Jul 11 2025: (Start)
Row n = 8 counts the following partitions by number of singleton parts other than the largest part:
  (8)                (5,3)        (4,3,1)
  (4,4)              (6,2)        (5,2,1)
  (4,2,2)            (7,1)
  (6,1,1)            (3,3,2)
  (2,2,2,2)          (3,2,2,1)
  (3,3,1,1)          (4,2,1,1)
  (5,1,1,1)          (3,2,1,1,1)
  (2,2,2,1,1)
  (4,1,1,1,1)
  (2,2,1,1,1,1)
  (3,1,1,1,1,1)
  (2,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1)
(End)

Alois P. Heinz, Rows n = 1..800, flattened

Row sums are A000041.
Row lengths are A003056.
For distinct parts instead of anti-runs we have A116608.
Column k = 1 is A116931.
For runs instead of anti-runs we have A384881.
The strict case is A384905.
The corresponding rank statistic is A356228, non-strict version A384906.
The proper case is A385814, runs A385815.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
Cf. A000009, A008284, A024786, A047993, A098859, A116674, A287170, A325325, A384882, A385213.

g := add(x^j*mul(1+t*x^i+x^(2*i)/(1-x^i), i = 1 .. j-1)/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 27)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n to 25 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i, t) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-1, t or j>0)*
      `if`(t and j=1, x, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, false)):
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 13 2016

b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, t || j > 0]*If[t && j == 1, x, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, False]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1!=#2+1&]]==k&]],{n,0,10},{k,0,n}] (* Delete zeros for A268193. Gus Wiseman, Jul 10 2025 *)

T(n,0) = A116931(n).
Sum_{k>=1} T(n, k) = A000041(n) (the partition numbers).
Sum_{k>=1} k*T(n,k) = A024786(n-1).
G.f.: G(t,x) = Sum_{j>=1} ((x^j/(1-x^j))*Product_{i=1..j-1} (1 + tx^i + x^{2i}/(1-x^i))).

A116674 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k distinct parts (n>=1, k>=1).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 3, 1, 5, 3, 4, 1, 2, 7, 1, 2, 8, 2, 2, 10, 3, 2, 11, 5, 2, 13, 7, 4, 12, 11, 1, 19, 11, 1, 2, 18, 17, 1, 3, 20, 21, 2, 2, 22, 27, 3, 2, 25, 32, 5, 4, 24, 41, 7, 2, 30, 46, 11, 2, 31, 56, 15, 2, 36, 62, 22, 3, 33, 80, 25, 1, 2, 39, 87, 36, 1, 4, 38, 103, 45, 2, 2, 45

Offset: 1

Emeric Deutsch, Feb 22 2006

Row n has floor(sqrt(n)) terms. Row sums yield A000009. T(n,1)=A001227(n) (n>=1). Sum(k*T(n,k),k>=1)=A038348(n-1) (n>=1).

Conjecture: Also the number of strict integer partitions of n with k maximal runs of consecutive parts decreasing by 1. - Gus Wiseman, Jun 24 2025

			From _Gus Wiseman_, Jun 24 2025: (Start)
Triangle begins:
   1:  1
   2:  1
   3:  2
   4:  1  1
   5:  2  1
   6:  2  2
   7:  2  3
   8:  1  5
   9:  3  4  1
  10:  2  7  1
  11:  2  8  2
  12:  2 10  3
  13:  2 11  5
  14:  2 13  7
  15:  4 12 11
  16:  1 19 11  1
  17:  2 18 17  1
  18:  3 20 21  2
  19:  2 22 27  3
  20:  2 25 32  5
Row n = 9 counts the following partitions into odd parts by number of distinct parts:
  (9)                  (7,1,1)          (5,3,1)
  (3,3,3)              (3,3,1,1,1)
  (1,1,1,1,1,1,1,1,1)  (5,1,1,1,1)
                       (3,1,1,1,1,1,1)
Row n = 9 counts the following strict partitions by number of maximal runs:
  (9)      (6,3)    (5,3,1)
  (5,4)    (7,2)
  (4,3,2)  (8,1)
           (6,2,1)
(End)

Alois P. Heinz, Rows n = 1..1000, flattened

Row sums are A000009, strict case of A000041.
Row lengths are A000196.
Leading terms are A001227.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat partitions, ranks A066311 or A073491.
A047993 counts partitions with max part = length.
A152140 counts partitions into odd parts by length.
A268193 counts partitions by number of maximal anti-runs, strict A384905.
A384881 counts partitions by number of maximal runs.
Cf. A008284, A038348, A047966, A089259, A325325, A384178, A384886, A384887.

g:=product(1+t*x^(2*j-1)/(1-x^(2*j-1)),j=1..35): gser:=simplify(series(g,x=0,34)): for n from 1 to 29 do P[n]:=coeff(gser,x^n) od: for n from 1 to 29 do seq(coeff(P[n],t,j),j=1..floor(sqrt(n))) od; # yields sequence in triangular form
# second Maple program:
with(numtheory):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-2)*`if`(j=0, 1, x), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(
         b(n, iquo(n+1, 2)*2-1)):
seq(T(n), n=1..30);  # Alois P. Heinz, Mar 08 2015

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-2]*If[j == 0, 1, x], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, Quotient[n+1, 2]*2-1]]; Table[T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OddQ[Times@@#]&&Length[Union[#]]==k&]],{n,1,12},{k,1,Floor[Sqrt[n]]}] (*  Gus Wiseman, Jun 24 2025 *)

G.f.: product(1+tx^(2j-1)/(1-x^(2j-1)), j=1..infinity).

cp's OEIS Frontend

A064174 Number of partitions of n with nonnegative rank.

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A340653 Number of balanced factorizations of n.

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A326836 Heinz numbers of integer partitions whose maximum part divides their sum.

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A384175 Number of subsets of {1..n} with all distinct lengths of maximal runs (increasing by 1).

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A349156 Number of integer partitions of n whose mean is not an integer.

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A160786 The number of odd partitions of consecutive odd integers.

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A168659 Number of partitions of n such that the number of parts is divisible by the greatest part. Also number of partitions of n such that the greatest part is divisible by the number of parts.

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