A324516 -id:A324516 - cp's OEIS frontend

A047993 Number of balanced partitions of n: the largest part equals the number of parts.

1, 0, 1, 1, 1, 1, 3, 2, 4, 4, 6, 7, 11, 11, 16, 19, 25, 29, 40, 45, 60, 70, 89, 105, 134, 156, 196, 232, 285, 336, 414, 485, 591, 696, 839, 987, 1187, 1389, 1661, 1946, 2311, 2702, 3201, 3731, 4400, 5126, 6018, 6997, 8195, 9502, 11093, 12849, 14949, 17281, 20062

Offset: 1

Wouter Meeussen

Useful in the creation of plane partitions with C3 or C3v symmetry.
The function T[m,a,b] used here gives the partitions of m whose Ferrers plot fits within an a X b box.
Central terms of triangle in A063995: a(n) = A063995(n,0). - Reinhard Zumkeller, Jul 24 2013
Sequence enumerates the collection of partitions of size n that are in the monoid of Dyson rank=0, or balanced partitions, under the binary operation A*B = (a1,a2,...,a[k-1],k)*(b1,...,b[n-1,n) = (a1*b1,...,a1*n,a2*b1,...,a2*n,...,k*b1,...,k*n), where A is a partition with k parts and B is a partition with n parts, and A*B is a partition with k*n parts. Note that the rank of A*B is 0, as required. For example, the product of the rank 0 partitions (1,2,3) of 6 and (1,1,3) of 5 is the rank 0 partition (1,1,2,2,3,3,3,6,9) of 30. There is no rank zero partition of 2, as shown in the sequence. It can be seen that any element of the monoid that partitions an odd prime p or a composite number of form 2p cannot be a product of smaller nontrivial partitions, whether in this monoid or not. - Richard Locke Peterson, Jul 15 2018
The "multiplication" given above was noted earlier by Franklin T. Adams-Watters in A122697. - Richard Peterson, Jul 19 2023
The Heinz numbers of these integer partitions are given by A106529. - Gus Wiseman, Mar 09 2019

			From _Joerg Arndt_, Oct 08 2012: (Start)
a(12) = 7 because the partitions of 12 where the largest part equals the number of parts are
   2 + 3 + 3 + 4,
   2 + 2 + 4 + 4,
   1 + 3 + 4 + 4,
   1 + 2 + 2 + 2 + 5,
   1 + 1 + 2 + 3 + 5,
   1 + 1 + 1 + 4 + 5, and
   1 + 1 + 1 + 1 + 2 + 6.
(End)
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(13) = 11 integer partitions:
  1  21  22  311  321  322   332   333    4222   4322    4332    4333
                       331   4211  4221   4321   4331    4422    4432
                       4111        4311   4411   4421    4431    4441
                                   51111  52111  52211   52221   52222
                                                 53111   53211   53221
                                                 611111  54111   53311
                                                         621111  54211
                                                                 55111
                                                                 622111
                                                                 631111
                                                                 7111111
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Erich Friedman, Illustration of initial terms

Cf. A000700, A063995, A064173, A064174.
Cf. A003114, A006141, A039900, A090858, A106529, A324516, A324518, A324520.
Column 1 of A350879.

a047993 = flip a063995 0  -- Reinhard Zumkeller, Jul 24 2013

A047993 := 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(A047993(n),n=1..20) ; # Emeric Deutsch, Dec 11 2004

Table[ Count[Partitions[n], par_List/; First[par]===Length[par]], {n, 12}] or recur: Sum[T[n-(2m-1), m-1, m-1], {m, Ceiling[Sqrt[n]], Floor[(n+1)/2]}] with T[m_, a_, b_]/; b < a := T[m, b, a]; T[m_, a_, b_]/; m > a*b := 0; T[m_, a_, b_]/; (2m > a*b) := T[a*b-m, a, b]; T[m_, 1, b_] := If[b < m, 0, 1]; T[0, , ] := 1; T[m_, a_, b_] := T[m, a, b]=Sum[T[m-a*i, a-1, b-i], {i, 0, Floor[m/a]}];
Table[Sum[ -(-1)^k*(p[n-(3*k^2-k)/2] - p[n-(3*k^2+k)/2]), {k, 1, Floor[(1+Sqrt[1+24*n])/6]}] /. p -> PartitionsP, {n, 1, 64}] (* Wouter Meeussen *)
(* also *)
Table[Count[IntegerPartitions[n], q_ /; Max[q] == Length[q]], {n, 24}]
(* Clark Kimberling, Feb 13 2014 *)
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*x^2*(1 - x^(2*k - 1))*(1 + x^k)/(1 - x^k), {x, 0, nmax}]]; s += p;, {k, 1, nmax + 1}]; Take[CoefficientList[s, x], nmax] (* Vaclav Kotesovec, Oct 16 2024 *)

N=66;  q='q + O('q^N );
S=2+2*ceil(sqrt(N));
gf= sum(k=1, S,  (-1)^k * ( q^((3*k^2+k)/2) - q^((3*k^2-k)/2) ) ) / prod(k=1,N, 1-q^k );
/* Joerg Arndt, Oct 08 2012 */

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)*prod(j=1, k-1, (1-x^(k+j-1))/(1-x^j)))) \\ Seiichi Manyama, Jan 24 2022

a(n) = p(n-1) - p(n-2) - p(n-5) + p(n-7) + ... + (-1)^k*(p(n-(3*k^2-k)/2) - p(n-(3*k^2+k)/2)) + ..., where p() is A000041(). E.g., a(20) = p(19) - p(18) - p(15) + p(13) + p(8) - p(5) = 490 - 385 - 176 + 101 + 22 - 7 = 45. - Vladeta Jovovic, Aug 04 2004
G.f.: ( Sum_{k>=1} (-1)^k * ( x^((3*k^2+k)/2) - x^((3*k^2-k)/2) ) ) / Product_{k>=1} (1-x^k). - Vladeta Jovovic, Aug 05 2004
a(n) ~ exp(Pi*sqrt(2*n/3))*Pi / (48*sqrt(2)*n^(3/2)) ~ p(n) * Pi / (4*sqrt(6*n)), where p(n) is the partition function A000041. - Vaclav Kotesovec, Oct 06 2016
G.f.: Sum_{n>=0} [2n,n]q q^(2*n), where [m,n]_q are the q-binomial coefficients. - _Mamuka Jibladze, Aug 12 2021
G.f.: Sum_{k>=1} x^(2*k-1) * Product_{j=1..k-1} (1-x^(k+j-1))/(1-x^j). - Seiichi Manyama, Jan 24 2022

A006141 Number of integer partitions of n whose smallest part is equal to the number of parts.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 25, 29, 33, 38, 42, 49, 54, 62, 69, 78, 87, 99, 109, 123, 137, 154, 170, 191, 211, 236, 261, 290, 320, 357, 392, 435, 479, 530, 582, 644, 706, 779, 854, 940, 1029, 1133, 1237, 1358, 1485

Offset: 1

N. J. A. Sloane

nonn

Or, number of partitions of n in which number of largest parts is equal to the largest part.
a(n) is the number of partitions of n-1 without parts that differ by less than 2 and which have no parts less than three. [MacMahon]
There are two conflicting choices for the offset in this sequence. For the definition given here the offset is 1, and that is what we shall adopt. On the other hand, if one arrives at this sequence via the Rogers-Ramanujan identities (see the next comment), the natural offset is 0.
Related to Rogers-Ramanujan identities: Let G[1](q) and G[2](q) be the generating functions for the two Rogers-Ramanujan identities of A003114 and A003106, starting with the constant term 1. The g.f. for the present sequence is G[3](q) = (G[1](q) - G[2](q))/q = 1+q^3+q^4+q^5+q^6+q^7+2*q^8+2*q^9+3*q^10+.... - Joerg Arndt, Oct 08 2012; N. J. A. Sloane, Nov 18 2015
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[3](x). - N. J. A. Sloane, Nov 22 2015
From Wolfdieter Lang, Oct 31 2016: (Start)
From Hardy (H) p. 94, eq. (6.12.1) and Hardy-Wright (H-W), p. 293,  eq. (19.14.3) for H_2(a,x) - H_1(a,x) = a*H_1(a*x,x) one finds from the result for H_1(a,x) (in (H) on top on p. 95), after putting a=x, the o.g.f. of a(n) = A003114(n) - A003106(n), n >= 0, with a(0) = 0 as Sum_{m>=0} x^((m+1)^2) / Product_{j=1..m} (1 - x^j). The m=0 term is  1*x^1. See the formula given by Joerg Arndt, Jan 29 2011.
This formula has a combinatorial interpretation (found similar to the one given in (H) section 6.0, pp. 91-92 or (H-W) pp. 290-291): a(n) is the number of partitions of n with parts differing by at least 2 and part 1 present. See the example for a(15) below. (End)
The Heinz numbers of these integer partitions are given by A324522. - Gus Wiseman, Mar 09 2019

			G.f. = x + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 3*x^12 + ...
a(15) = 5 because the partitions of 15 where the smallest part equals the number of parts are
3 + 6 + 6,
3 + 5 + 7,
3 + 4 + 8,
3 + 3 + 9, and
2 + 13.
- _Joerg Arndt_, Oct 08 2012
a(15) = 5 because the partitions of 15 with parts differing by at least 2 and part 1 present are: [14,1] obtained from the partition of 11 with one part, [11], added to the first part of the special partition [3,1] of 4 and  [11,3,1], [10,4,1], [9,5,1], [8,6,1] from adding all partition of 15 - 9 = 6 with one part, [6], and those with two parts, [5,1], [4,1], [3,3], to the special partition [5,3,1] of 9. - _Wolfdieter Lang_, Oct 31 2016
a(15) = 5 because the partitions of 14 with parts >= 3 and parts differing by at least 2 are [14], [11,3], [10,4], [9,5] and [8,6]. See the second [MacMahon] comment. This follows from the g.f. G[3](q) given in Andrews - Baxter, eq. (5.1) for i=3, (using summation index  m) and  m*(m+2) = 3 + 5 + ... + (2*m+1). - _Wolfdieter Lang_, Nov 02 2016
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(8) = 1 through a(15) = 5 integer partitions:
  (6,2)  (7,2)    (8,2)    (9,2)    (10,2)   (11,2)   (12,2)   (13,2)
         (3,3,3)  (4,3,3)  (4,4,3)  (5,4,3)  (5,5,3)  (6,5,3)  (6,6,3)
                           (5,3,3)  (6,3,3)  (6,4,3)  (7,4,3)  (7,5,3)
                                             (7,3,3)  (8,3,3)  (8,4,3)
                                                               (9,3,3)
(End)

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 92-95.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 292-294.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 45, Section 293.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
George E. Andrews and R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Handout, Math. Dept., Rutgers University, April 2015.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Dissertation, Math. Dept., Rutgers University, April 2015.
James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, arXiv:1205.6570 [math.CO], 2012; The Ramanujan Journal 29.1-3 (2012): 199-211.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
Cf. A064174, A090858, A324516, A324518, A324520, A324522.
A003106 counts partitions with minimum > length.
A003114 counts partitions with minimum >= length.
A026794 counts partitions by minimum.
A039899 counts partitions with minimum < length.
A039900 counts partitions with minimum <= length.
A239950 counts partitions with minimum equal to number of distinct parts.
Sequences related to balance:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 ranks balanced partitions.
- A340596 counts co-balanced factorizations.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
Cf. A055396, A114638, A117409, A325134, A340601, A340611, A340654, A340655.

b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i,i)))))
    end:
a:= n-> add(b(n-j^2, j-1), j=0..isqrt(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 08 2012

b[n_, i_] := b[n, i] = If[n<0, 0, If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]]; a[n_] := Sum[b[n-j^2, j-1], {j, 0, Sqrt[n]}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Min[#]==Length[#]&]],{n,30}] (* Gus Wiseman, Mar 09 2019 *)

{a(n) = if( n<1, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(j=1, k-1, 1 - x^j, 1 + O(x ^ (n - k^2 + 1) ))), n))} /* Michael Somos, Jan 22 2008 */

G.f.: Sum_{m>=1} (x^(m^2)-x^(m*(m+1))) / Product_{i=1..m} (1-x^i) .
G.f.: Sum_{n>=1} x^(n^2)/Product_{k=1..n-1} (1-x^k). - Joerg Arndt, Jan 29 2011
a(n) = A003114(n) - A003106(n) = A039900(n) - A039899(n), (offset 1). - Vladeta Jovovic, Jul 17 2004
Plouffe in his 1992 dissertation conjectured that this has g.f. = (1+z+z^4+2*z^5-z^3-z^8+3*z^10-z^7+z^9)/(1+z-z^4-2*z^3-z^8+z^10), but Michael Somos pointed out on Jan 22 2008 that this is false.
Expansion of ( f(-x^2, -x^3) - f(-x, -x^4) ) / f(-x) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jan 22 2007
a(n) ~ sqrt(1/sqrt(5) - 2/5) * exp(2*Pi*sqrt(n/15)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Nov 01 2016

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000
Better description from Naohiro Nomoto, Feb 06 2002
Name shortened by Gus Wiseman, Apr 07 2021 (balanced partitions are A047993).

A064174 Number of partitions of n with nonnegative rank.

Original entry on oeis.org

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

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

A340601 Number of integer partitions of n of even rank.

Original entry on oeis.org

1, 1, 0, 3, 1, 5, 3, 11, 8, 18, 16, 34, 33, 57, 59, 98, 105, 159, 179, 262, 297, 414, 478, 653, 761, 1008, 1184, 1544, 1818, 2327, 2750, 3480, 4113, 5137, 6078, 7527, 8899, 10917, 12897, 15715, 18538, 22431, 26430, 31805, 37403, 44766, 52556, 62620, 73379

Offset: 0

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. For this sequence, the rank of an empty partition is 0.

			The a(1) = 1 through a(9) = 18 partitions (empty column indicated by dot):
  (1)  .  (3)    (22)  (5)      (42)    (7)        (44)      (9)
          (21)         (41)     (321)   (43)       (62)      (63)
          (111)        (311)    (2211)  (61)       (332)     (81)
                       (2111)           (322)      (521)     (333)
                       (11111)          (331)      (2222)    (522)
                                        (511)      (4211)    (531)
                                        (2221)     (32111)   (711)
                                        (4111)     (221111)  (4221)
                                        (31111)              (4311)
                                        (211111)             (6111)
                                        (1111111)            (32211)
                                                             (33111)
                                                             (51111)
                                                             (222111)
                                                             (411111)
                                                             (3111111)
                                                             (21111111)
                                                             (111111111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
The positive case is A101708 (A340605).
The Heinz numbers of these partitions are A340602.
The odd version is A340692 (A340603).
- Rank -
A047993 counts partitions of rank 0 (A106529).
A072233 counts partitions by sum and length.
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts factorizations of rank 0.
- Even -
A024430 counts set partitions of even length.
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A052841 counts ordered set partitions of even length.
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts even-length partitions of even numbers (A340784).
A339846 counts factorizations of even length.
Cf. A000041, A006141, A039900, A064174, A067538, A117409, A200750, A324516.

b:= proc(n, i, r) option remember; `if`(n=0, 1-max(0, r),
      `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
      `if`(r<0, irem(i, 2), r))))
    end:
a:= n-> b(n$2, -1):
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 22 2021

Table[If[n==0,1,Length[Select[IntegerPartitions[n],EvenQ[Max[#]-Length[#]]&]]],{n,0,30}]
(* Second program: *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, 1 - Max[0, r], If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 - If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1];
a /@ Range[0, 55] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

p_q(k) = {prod(j=1, k, 1-q^j); }
GB_q(N, M)= {if(N>=0 && M>=0,  p_q(N+M)/(p_q(M)*p_q(N)), 0 ); }
A_q(N) = {my(q='q+O('q^N), g=1+sum(i=1,N, sum(j=1,N/i, q^(i*j) * ( ((1/2)*(1+(-1)^(i+j))) + sum(k=1,N-(i*j), ((q^k)*GB_q(k,i-2)) * ((1/2)*(1+(-1)^(i+j+k)))))))); Vec(g)}
A_q(50) \\ John Tyler Rascoe, Apr 15 2024

G.f.: 1 + Sum_{i, j>0} q^(i*j) * ( (1+(-1)^(i+j))/2 + Sum_{k>0} q^k * q_binomial(k,i-2) * (1+(-1)^(i+j+k))/2 ). - John Tyler Rascoe, Apr 15 2024

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - Vaclav Kotesovec, Apr 17 2024

A114638 Number of partitions of n such that number of parts is equal to the sum of parts counted without multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 1, 0, 2, 2, 3, 5, 5, 6, 9, 7, 8, 14, 12, 16, 21, 28, 32, 43, 47, 61, 68, 84, 89, 109, 126, 140, 170, 198, 227, 261, 323, 362, 427, 501, 581, 658, 794, 880, 1036, 1175, 1355, 1526, 1776, 1985, 2281, 2588, 2943, 3312, 3799, 4271, 4852, 5497

Offset: 0

Vladeta Jovovic, Feb 18 2006

nonn

The Heinz numbers of these integer partitions are given by A324570. - Gus Wiseman, Mar 09 2019

			a(10) = 3 because we have [5,1,1,1,1,1], [3,3,3,1] and [3,2,2,1,1,1].
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(12) = 5 integer partitions (empty columns not shown):
  1  22   221  3111  3311   333     3331    32222    33222
     211             41111  321111  322111  44111    322221
                                    511111  322211   332211
                                            332111   4221111
                                            4211111  6111111
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A003114, A006141, A039900, A047993, A064174, A066328, A243149 (the same for compositions).
Cf. A324516, A324518, A324520, A324521, A324570.
Cf. A116861 (number of partitions of n having a given sum of distinct parts).

a:=proc(n) local P,c,j,S: with(combinat): P:=partition(n): c:=0: for j from 1 to nops(P) do S:=convert(P[j],set): if nops(P[j])=sum(S[i],i=1..nops(S)) then c:=c+1 else c:=c fi: c: od: end: seq(a(n), n=0..35); # Emeric Deutsch, Mar 01 2006

a[n_] := Module[{P, c, j, S}, P = IntegerPartitions[n]; c = 0; For[j = 1, j <= Length[P], j++, S = Union[P[[j]]]; If[Length[P[[j]]] == Total[S],  c++] ]; c];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 07 2018, after Emeric Deutsch *)

apply( A114638(n,s=0)={forpart(p=n,#p==vecsum(Set(p))&&s++); s}, [0..50]) \\ M. F. Hasler, Oct 27 2019

More terms from Emeric Deutsch, Mar 01 2006

A324518 Number of integer partitions of n > 0 where the maximum part equals the length minus the number of distinct parts.

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 2, 0, 3, 1, 6, 7, 7, 9, 11, 10, 16, 26, 22, 42, 43, 54, 61, 83, 85, 118, 135, 179, 201, 263, 297, 371, 445, 510, 608, 732, 886, 1009, 1231, 1442, 1721, 2015, 2416, 2750, 3327, 3784, 4542, 5190, 6142, 7044, 8315, 9573, 11203, 12913, 15056

Offset: 1

Gus Wiseman, Mar 06 2019

nonn

The Heinz numbers of these integer partitions are given by A324517.

			The a(2) = 1 through a(12) = 7 integer partitions:
  (11)  (2111)  (222)   (2221)   (33111)   (322111)  (32222)    (3333)
                (2211)  (31111)  (321111)            (33311)    (33222)
                                 (411111)            (322211)   (322221)
                                                     (332111)   (332211)
                                                     (4211111)  (441111)
                                                     (5111111)  (4221111)
                                                                (4311111)

Cf. A003114, A006141, A039900, A046660, A047993, A064174, A106529.

Cf. A324516, A324517, A324520, A324562.

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

A340602 Heinz numbers of integer partitions of even rank.

Original entry on oeis.org

1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 35, 36, 38, 39, 41, 44, 45, 47, 49, 50, 54, 56, 57, 58, 59, 65, 66, 67, 68, 73, 74, 75, 80, 81, 83, 84, 86, 87, 91, 92, 95, 96, 97, 99, 102, 103, 104, 106, 109, 110, 111, 120, 122, 124, 125, 126, 127

Offset: 1

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is 0.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of partitions with their Heinz numbers begins:
     1: ()           31: (11)           58: (10,1)
     2: (1)          32: (1,1,1,1,1)    59: (17)
     5: (3)          35: (4,3)          65: (6,3)
     6: (2,1)        36: (2,2,1,1)      66: (5,2,1)
     8: (1,1,1)      38: (8,1)          67: (19)
     9: (2,2)        39: (6,2)          68: (7,1,1)
    11: (5)          41: (13)           73: (21)
    14: (4,1)        44: (5,1,1)        74: (12,1)
    17: (7)          45: (3,2,2)        75: (3,3,2)
    20: (3,1,1)      47: (15)           80: (3,1,1,1,1)
    21: (4,2)        49: (4,4)          81: (2,2,2,2)
    23: (9)          50: (3,3,1)        83: (23)
    24: (2,1,1,1)    54: (2,2,2,1)      84: (4,2,1,1)
    26: (6,1)        56: (4,1,1,1)      86: (14,1)
    30: (3,2,1)      57: (8,2)          87: (10,2)

FindStat, St000145: The Dyson rank of a partition

Taking only length gives A001222.
Taking only maximum part gives A061395.
These partitions are counted by A340601.
The complement is A340603.
The case of positive rank is A340605.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A324516 counts partitions with rank = maximum minus minimum part (A324515).
A340653 counts factorizations of rank 0.
A340692 counts partitions of odd rank (A340603).
- Even -
A024430 counts set partitions of even length.
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A052841 counts ordered set partitions of even length.
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts even-length partitions of even numbers (A340784).
A339846 counts factorizations of even length.
Cf. A000041, A006141, A056239, A072233, A112798, A168659, A325134, A326836, A326845, A340386, A340387.

Select[Range[100],EvenQ[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]&]

Either n = 1 or A061395(n) - A001222(n) is even.

A324520 Number of integer partitions of n > 0 where the minimum part equals the number of parts minus the number of distinct parts.

Original entry on oeis.org

0, 1, 0, 1, 2, 2, 3, 3, 7, 6, 11, 12, 15, 21, 25, 31, 43, 49, 58, 79, 89, 108, 135, 165, 190, 232, 279, 328, 387, 461, 536, 650, 743, 870, 1029, 1202, 1381, 1613, 1864, 2163, 2505, 2875, 3292, 3829, 4367, 5001, 5746, 6538, 7462, 8533, 9714, 11008, 12527, 14196

Offset: 1

Gus Wiseman, Mar 06 2019

nonn

The Heinz numbers of these integer partitions are given by A324519.

			The a(2) = 1 through a(11) = 11 integer partitions:
  (11)  (211)  (221)  (222)  (331)   (611)   (441)   (811)   (551)
               (311)  (411)  (511)   (3221)  (711)   (3322)  (911)
                             (3211)  (4211)  (3222)  (4222)  (3332)
                                             (3321)  (5221)  (4331)
                                             (4221)  (5311)  (4421)
                                             (4311)  (6211)  (5222)
                                             (5211)          (5411)
                                                             (6221)
                                                             (6311)
                                                             (7211)
                                                             (43211)

Cf. A003114, A006141, A039900, A046660, A047993, A064174, A090858, A133121.

Cf. A324516, A324518, A324519, A324520.

Table[Length[Select[IntegerPartitions[n],Min@@#==Length[#]-Length[Union[#]]&]],{n,30}]

A324515 Numbers > 1 where the maximum prime index minus the minimum prime index equals the number of prime factors minus the number of distinct prime factors.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 29, 31, 37, 40, 41, 43, 45, 47, 53, 59, 61, 67, 71, 73, 75, 79, 83, 89, 97, 100, 101, 103, 107, 109, 112, 113, 120, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 175, 179, 180, 181, 189, 191, 193, 197, 199, 211, 223

Offset: 1

Gus Wiseman, Mar 06 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Also Heinz numbers of the integer partitions enumerated by A324516. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  12: {1,1,2}
  13: {6}
  17: {7}
  18: {1,2,2}
  19: {8}
  23: {9}
  29: {10}
  31: {11}
  37: {12}
  40: {1,1,1,3}
  41: {13}
  43: {14}
  45: {2,2,3}

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A006141, A046660, A047993, A055396, A056239, A061395, A106529, A112798, A243055.

Cf. A324516, A324517, A324519, A324521, A324522, A324560, A324562.

filter:= proc(n) local F, Inds, t;
  if isprime(n) then return true fi;
  F:= ifactors(n)[2];
  Inds:= map(numtheory:-pi, F[..,1]);
  max(Inds) - min(Inds) = add(t[2],t=F) - nops(F)
end proc:
select(filter, [$2..300]); # Robert Israel, Nov 19 2023

Select[Range[2,100],With[{f=FactorInteger[#]},PrimePi[f[[-1,1]]]-PrimePi[f[[1,1]]]==Total[Last/@f]-Length[f]]&]

A243055(a(n)) = A061395(a(n)) - A055396(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).

cp's OEIS Frontend

A047993 Number of balanced partitions of n: the largest part equals the number of parts.

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A006141 Number of integer partitions of n whose smallest part is equal to the number of parts.

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A064174 Number of partitions of n with nonnegative rank.

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A064173 Number of partitions of n with positive rank.

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A340601 Number of integer partitions of n of even rank.

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A114638 Number of partitions of n such that number of parts is equal to the sum of parts counted without multiplicities.

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