A064391 -id:A064391 - cp's OEIS frontend

A001522 Number of n-stacks with strictly receding walls, or the number of Type A partitions of n in the sense of Auluck (1951).

1, 1, 1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 35, 47, 62, 82, 107, 139, 179, 230, 293, 372, 470, 591, 740, 924, 1148, 1422, 1756, 2161, 2651, 3244, 3957, 4815, 5844, 7075, 8545, 10299, 12383, 14859, 17794, 21267, 25368, 30207, 35902, 42600, 50462, 59678, 70465, 83079, 97800, 114967, 134956, 158205, 185209, 216546, 252859

Offset: 0

N. J. A. Sloane

Also number of partitions of n with positive crank (n>=2), cf. A064391. - Vladeta Jovovic, Sep 30 2001
Number of smooth weakly unimodal compositions of n into positive parts such that the first and last part are 1 (smooth means that successive parts differ by at most one), see example. Dropping the requirement for unimodality gives A186085. - Joerg Arndt, Dec 09 2012
Number of weakly unimodal compositions of n where the maximal part m appears at least m times, see example. - Joerg Arndt, Jun 11 2013
Also weakly unimodal compositions of n with first part 1, maximal up-step 1, and no consecutive up-steps; see example. The smooth weakly unimodal compositions are recovered by shifting all rows above the bottom row to the left by one position with respect to the next lower row. - Joerg Arndt, Mar 30 2014
It would seem from Stanley that he regards a(0)=0 for this sequence and A001523. - Michael Somos, Feb 22 2015
From Gus Wiseman, Mar 30 2021: (Start)
Also the number of odd-length compositions of n with alternating parts strictly decreasing. These are finite odd-length sequences q of positive integers summing to n such that q(i) > q(i+2) for all possible i. The even-length version is A064428. For example, the a(1) = 1 through a(9) = 14 compositions are:
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)
                 (211)  (221)  (231)  (241)  (251)  (261)
                        (311)  (312)  (322)  (332)  (342)
                               (321)  (331)  (341)  (351)
                               (411)  (412)  (413)  (423)
                                      (421)  (422)  (432)
                                      (511)  (431)  (441)
                                             (512)  (513)
                                             (521)  (522)
                                             (611)  (531)
                                                    (612)
                                                    (621)
                                                    (711)
                                                    (32211)
(End)
In the Ferrers diagram of a partition x of n, count the dots in each diagonal parallel to the main diagonal (starting at the top-right, say). The result diag(x) is a smooth weakly unimodal composition of n into positive parts such that the first and last part are 1. For example, diag(5541) = 11233221. The function diag is many-to-one; the size of its codomain as a set is a(n). If diag(x) = diag(y), each hook of x can be slid by the same amount past the main diagonal to get y. For example, diag(5541) = diag(44331). - George Beck, Sep 26 2021
From Gus Wiseman, May 23 2022: (Start)
Conjecture: Also the number of integer partitions y of n with a fixed point y(i) = i. These partitions are ranked by A352827. The conjecture is stated at A238395, but Resta tells me he may not have had a proof. The a(1) = 1 through a(8) = 10 partitions are:
  (1)  (11)  (111)  (22)    (32)     (42)      (52)       (62)
                    (1111)  (221)    (222)     (322)      (422)
                            (11111)  (321)     (421)      (521)
                                     (2211)    (2221)     (2222)
                                     (111111)  (3211)     (3221)
                                               (22111)    (4211)
                                               (1111111)  (22211)
                                                          (32111)
                                                          (221111)
                                                          (11111111)
Note that these are not the same partitions (compare A352827 to A352874), only the same count (apparently).
(End)
The above conjecture is true. See Section 4 of the Blecher-Knopfmacher paper in the Links section. - Jeremy Lovejoy, Sep 26 2022

			For a(6)=5 we have the following stacks:
.x... ..x.. ...x. .xx.
xxxxx xxxxx xxxxx xxxx xxxxxx
.
From _Joerg Arndt_, Dec 09 2012: (Start)
There are a(9) = 14 smooth weakly unimodal compositions of 9:
01:   [ 1 1 1 1 1 1 1 1 1 ]
02:   [ 1 1 1 1 1 1 2 1 ]
03:   [ 1 1 1 1 1 2 1 1 ]
04:   [ 1 1 1 1 2 1 1 1 ]
05:   [ 1 1 1 1 2 2 1 ]
06:   [ 1 1 1 2 1 1 1 1 ]
07:   [ 1 1 1 2 2 1 1 ]
08:   [ 1 1 2 1 1 1 1 1 ]
09:   [ 1 1 2 2 1 1 1 ]
10:   [ 1 1 2 2 2 1 ]
11:   [ 1 2 1 1 1 1 1 1 ]
12:   [ 1 2 2 1 1 1 1 ]
13:   [ 1 2 2 2 1 1 ]
14:   [ 1 2 3 2 1 ]
(End)
From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(9) = 14 weakly unimodal compositions of 9 where the maximal part m appears at least m times:
01:  [ 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 2 ]
03:  [ 1 1 1 1 2 2 1 ]
04:  [ 1 1 1 2 2 1 1 ]
05:  [ 1 1 1 2 2 2 ]
06:  [ 1 1 2 2 1 1 1 ]
07:  [ 1 1 2 2 2 1 ]
08:  [ 1 2 2 1 1 1 1 ]
09:  [ 1 2 2 2 1 1 ]
10:  [ 1 2 2 2 2 ]
11:  [ 2 2 1 1 1 1 1 ]
12:  [ 2 2 2 1 1 1 ]
13:  [ 2 2 2 2 1 ]
14:  [ 3 3 3 ]
(End)
From _Joerg Arndt_, Mar 30 2014: (Start)
There are a(9) = 14 compositions of 9 with first part 1, maximal up-step 1, and no consecutive up-steps:
01:  [ 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 1 2 1 1 ]
05:  [ 1 1 1 1 1 2 2 ]
06:  [ 1 1 1 1 2 1 1 1 ]
07:  [ 1 1 1 1 2 2 1 ]
08:  [ 1 1 1 2 1 1 1 1 ]
09:  [ 1 1 1 2 2 1 1 ]
10:  [ 1 1 1 2 2 2 ]
11:  [ 1 1 2 1 1 1 1 1 ]
12:  [ 1 1 2 2 1 1 1 ]
13:  [ 1 1 2 2 2 1 ]
14:  [ 1 1 2 2 3 ]
(End)
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 14*x^9 + ...

G. E. Andrews, The reasonable and unreasonable effectiveness of number theory in statistical mechanics, pp. 21-34 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
G. E. Andrews, Three-quadrant Ferrers graphs, Indian J. Math., 42 (No. 1, 2000), 1-7.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see section 2.5 on page 76.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)
A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Erich Friedman, Illustration of initial terms
A. D. Sokal, The leading root of the partial theta function, arXiv preprint arXiv:1106.1003 [math.CO], 2011.
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

Cf. A000041, A059776, A001523, A001524.
A version for permutations is A002467, complement A000166.
The case of zero crank is A064410, ranked by A342192.
The case of nonnegative crank is A064428, ranked by A352873.
A strict version is A352829, complement A352828.
Conjectured to be column k = 1 of A352833.
These partitions (positive crank) are ranked by A352874.
A000700 counts self-conjugate partitions, ranked by A088902.
A064391 counts partitions by crank.
A115720 and A115994 count partitions by their Durfee square.
A257989 gives the crank of the partition with Heinz number n.
Counting compositions: A003242, A114921, A238351, A342527, A342528, A342532.
Fixed points of reversed partitions: A238352, A238394, A238395, A352822, A352830, A352872.
Cf. A008292, A114088, A118199, A325187, A352827, A352832.

b:= proc(n, i, t) option remember; `if`(n<=0, `if`(i=1, 1, 0),
      `if`(n<0 or i<1, 0, b(n-i, i, t)+b(n-(i-1), i-1, false)+
      `if`(t, b(n-(i+1), i+1, t), 0)))
    end:
a:= n-> b(n-1, 1, true):
seq(a(n), n=0..70);  # Alois P. Heinz, Feb 26 2014
# second Maple program:
A001522 := proc(n)
    local r,a;
    a := 0 ;
    if n = 0 then
        return 1 ;
    end if;
    for r from 1 do
        if r*(r+1) > 2*n then
            return a;
        else
            a := a-(-1)^r*combinat[numbpart](n-r*(r+1)/2) ;
        end if;
    end do:
end proc: # R. J. Mathar, Mar 07 2015

max = 50; f[x_] := 1 + Sum[-(-1)^k*x^(k*(k+1)/2), {k, 1, max}] / Product[(1-x^k), {k, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 27 2011, after g.f. *)
b[n_, i_, t_] := b[n, i, t] = If[n <= 0, If[i == 1, 1, 0], If[n<0 || i<1, 0, b[n-i, i, t] + b[n - (i-1), i-1, False] + If[t, b[n - (i+1), i+1, t], 0]]]; a[n_] := b[n-1, 1, True]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Dec 01 2015, after Alois P. Heinz *)
Flatten[{1, Table[Sum[(-1)^(j-1)*PartitionsP[n-j*((j+1)/2)], {j, 1, Floor[(Sqrt[8*n + 1] - 1)/2]}], {n, 1, 60}]}] (* Vaclav Kotesovec, Sep 26 2016 *)
ici[q_]:=And@@Table[q[[i]]>q[[i+2]],{i,Length[q]-2}];
Table[If[n==0,1,Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],OddQ@*Length],ici]]],{n,0,15}] (* Gus Wiseman, Mar 30 2021 *)

{a(n) = if( n<1, n==0, polcoeff( sum(k=1, (sqrt(1+8*n) - 1)\2, -(-1)^k * x^((k + k^2)/2)) / eta(x + x * O(x^n)), n))}; /* Michael Somos, Jul 22 2003 */

N=66; q='q+O('q^N);
Vec( 1 + sum(n=1, N, q^(n^2)/(prod(k=1,n-1,1-q^k)^2*(1-q^n)) ) ) \\ Joerg Arndt, Dec 09 2012

def A001522(n):
    if n < 4: return 1
    return (number_of_partitions(n) - [p.crank() for p in Partitions(n)].count(0))/2
[A001522(n) for n in range(30)]  # Peter Luschny, Sep 15 2014

a(n) = (A000041(n) - A064410(n)) / 2 for n>=2.
G.f.: 1 + ( Sum_{k>=1} -(-1)^k * x^(k*(k+1)/2) ) / ( Product_{k>=1} 1-x^k ).
G.f.: 1 + ( Sum_{n>=1} q^(n^2) / ( ( Product_{k=1..n-1} 1-q^k )^2 * (1-q^n) ) ). - Joerg Arndt, Dec 09 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) [Auluck, 1951]. - Vaclav Kotesovec, Sep 26 2016
a(n) = A000041(n) - A064428(n). - Gus Wiseman, Mar 30 2021
a(n) = A064428(n) - A064410(n). - Gus Wiseman, May 23 2022

a(0) changed from 0 to 1 by Joerg Arndt, Mar 30 2014

Edited definition. - N. J. A. Sloane, Mar 31 2021

A064428 Number of partitions of n with nonnegative crank.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 6, 8, 12, 16, 23, 30, 42, 54, 73, 94, 124, 158, 206, 260, 334, 420, 532, 664, 835, 1034, 1288, 1588, 1962, 2404, 2953, 3598, 4392, 5328, 6466, 7808, 9432, 11338, 13632, 16326, 19544, 23316, 27806, 33054, 39273, 46534, 55096, 65076, 76808

Offset: 0

Vladeta Jovovic, Sep 30 2001

nonn

For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).
From Gus Wiseman, Mar 30 2021 and May 21 2022: (Start)
Also the number of even-length compositions of n with alternating parts strictly decreasing, or properly 2-colored partitions (proper = no equal parts of the same color) with the same number of parts of each color, or ordered pairs of strict partitions of the same length with total n. The odd-length case is A001522, and there are a total of A000041 compositions with alternating parts strictly decreasing (see A342528 for a bijective proof). The a(2) = 1 through a(7) = 8 ordered pairs of strict partitions of the same length are:
  (1)(1)  (1)(2)  (1)(3)  (1)(4)   (1)(5)    (1)(6)
          (2)(1)  (2)(2)  (2)(3)   (2)(4)    (2)(5)
                  (3)(1)  (3)(2)   (3)(3)    (3)(4)
                          (4)(1)   (4)(2)    (4)(3)
                                   (5)(1)    (5)(2)
                                  (21)(21)   (6)(1)
                                            (21)(31)
                                            (31)(21)
Conjecture: Also the number of integer partitions y of n without a fixed point y(i) = i, ranked by A352826. This is stated at A238394, but Resta tells me he may not have had a proof. The a(2) = 1 through a(7) = 8 partitions without a fixed point are:
  (2)  (3)   (4)    (5)     (6)      (7)
       (21)  (31)   (41)    (33)     (43)
             (211)  (311)   (51)     (61)
                    (2111)  (411)    (331)
                            (3111)   (511)
                            (21111)  (4111)
                                     (31111)
                                     (211111)
The version for permutations is A000166, complement A002467.
The version for compositions is A238351.
This is column k = 0 of A352833.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872. (End)
The above conjecture is true. See Section 4 of the Blecher-Knopfmacher paper in the Links section. - Jeremy Lovejoy, Sep 26 2022

			G.f. = 1 + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 12*x^8 + 16*x^9 + 23*x^10 + ... - _Michael Somos_, Jan 15 2018
From _Gus Wiseman_, May 21 2022: (Start)
The a(0) = 1 through a(8) = 12 partitions with nonnegative crank:
  ()  .  (2)  (3)   (4)   (5)    (6)    (7)     (8)
              (21)  (22)  (32)   (33)   (43)    (44)
                    (31)  (41)   (42)   (52)    (53)
                          (221)  (51)   (61)    (62)
                                 (222)  (322)   (71)
                                 (321)  (331)   (332)
                                        (421)   (422)
                                        (2221)  (431)
                                                (521)
                                                (2222)
                                                (3221)
                                                (3311)
(End)

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 18 Entry 9 Corollary (i).
G. E. Andrews, B. C. Berndt, Ramanujan's Lost Notebook Part I, Springer, see p. 169 Entry 6.7.1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
George E. Andrews and David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
Cody Armond and Oliver T. Dasbach, Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial, arXiv:1106.3948 [math.GT], 2011.
Cristina Ballantine and Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.
Rupam Barman and Ajit Singh, On Mex-related partition functions of Andrews and Newman, arXiv:2009.11602 [math.NT], 2020.
Aubrey Blecher and Arnold Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.
Brian Hopkins, James A. Sellers, and Ae Ja Yee, Combinatorial Perspectives on the Crank and Mex Partition Statistics, arXiv:2108.09414 [math.CO], 2021.
Mbavhalelo Mulokwe and Konstantinos Zoubos, Free fermions, neutrality and modular transformations, arXiv:2403.08531 [hep-th], 2024.

These are the row-sums of the right (or left) half of A064391, inclusive.
The case of crank 0 is A064410, ranked by A342192.
The strict case is A352828.
These partitions are ranked by A352873.
A000700 = self-conjugate partitions, ranked by A088902, complement A330644.
A001522 counts partitions with positive crank, ranked by A352874.
A034008 counts even-length compositions.
A115720 and A115994 count partitions by their Durfee square.
A224958 counts compositions w/ alternating parts unequal (even: A342532).
A257989 gives the crank of the partition with Heinz number n.
A342527 counts compositions w/ alternating parts equal (even: A065608).
A342528 = compositions w/ alternating parts weakly decr. (even: A114921).
Cf. A000041, A008292, A062968, A118199, A188674, A325547, A325548.

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^(k (k + 1)/2) , {k, 0, (Sqrt[1 + 8 n] - 1)/2}] / QPochhammer[ x], {x, 0, n}]]; (* Michael Somos, Jan 15 2018 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[  x^(k (k + 1)) / QPochhammer[ x, x, k]^2 , {k, 0, (Sqrt[1 + 4 n] - 1)/2}], {x, 0, n}]]; (* Michael Somos, Jan 15 2018 *)
ck[y_]:=With[{w=Count[y,1]},If[w==0,If[y=={},0,Max@@y],Count[y,?(#>w&)]-w]];Table[Length[Select[IntegerPartitions[n],ck[#]>=0&]],{n,0,30}] (* _Gus Wiseman, Mar 30 2021 *)
ici[q_]:=And@@Table[q[[i]]>q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], EvenQ@*Length],ici]],{n,0,15}] (* Gus Wiseman, Mar 30 2021 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(1 + 8*n) -1)\2, (-1)^k * x^((k+k^2)/2)) / eta( x + x * O(x^n)), n))}; /* Michael Somos, Jul 28 2003 */

a(n) = (A000041(n) + A064410(n)) / 2, n>1. - Michael Somos, Jul 28 2003
G.f.: (Sum_{k>=0} (-1)^k * x^(k(k+1)/2)) / (Product_{k>0} 1-x^k). - Michael Somos, Jul 28 2003
G.f.: Sum_{i>=0} x^(i*(i+1)) / (Product_{j=1..i} 1-x^j )^2. - Jon Perry, Jul 18 2004
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - Vaclav Kotesovec, Sep 26 2016
G.f.: (Sum_{i>=0} x^i / (Product_{j=1..i} 1-x^j)^2 ) * (Product_{k>0} 1-x^k). - Li Han, May 23 2020
a(n) = A000041(n) - A001522(n). - Gus Wiseman, Mar 30 2021
a(n) = A064410(n) + A001522(n). - Gus Wiseman, May 21 2022

A064410 Number of partitions of n with zero crank.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 2, 4, 4, 7, 7, 11, 12, 17, 19, 27, 30, 41, 48, 62, 73, 95, 110, 140, 166, 206, 243, 302, 354, 435, 513, 622, 733, 887, 1039, 1249, 1467, 1750, 2049, 2438, 2847, 3371, 3934, 4634, 5398, 6343, 7367, 8626, 10009, 11677, 13521, 15737, 18184

Offset: 1

Vladeta Jovovic, Sep 29 2001

For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).

			a(10)=4 because there are 4 partitions of 10 with zero crank: 1+1+2+3+3, 1+1+4+4, 1+1+3+5 and 1+9.
From _Gus Wiseman_, Apr 02 2021: (Start)
The a(3) = 1 through a(14) = 11 partitions (A..D = 10..13):
  21  31  41  51  61  71    81    91     A1     B1      C1      D1
                      3311  4311  4411   5411   5511    6511    6611
                                  5311   6311   6411    7411    7511
                                  33211  43211  7311    8311    8411
                                                44211   54211   9311
                                                53211   63211   55211
                                                332211  432211  64211
                                                                73211
                                                                442211
                                                                532211
                                                                3322211
(End)

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Brian Hopkins and James A. Sellers, On Blecher and Knopfmacher's Fixed Points for Integer Partitions, arXiv:2305.05096 [math.CO], 2023. Mentions this sequence.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020. Mentions this sequence.

The version for positive crank is A001522.
Central column of A064391.
The version for nonnegative crank is A064428.
The Heinz numbers of these partitions are A342192.
A003242 counts anti-run compositions.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
Cf. A000009, A000041, A000726, A027193, A034008, A062968, A114921, A188674, A342194, A342343, A342528.

nmax = 60; Rest[CoefficientList[Series[x - 1 + Sum[(-1)^k*(x^(k*(k + 1)/2) - x^(k*(k - 1)/2)), {k, 1, nmax}] / Product[1 - x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 26 2016 *)
Flatten[{0, Table[PartitionsP[n] - 2*Sum[(-1)^(j+1)*PartitionsP[n - j*((j+1)/2)], {j, 1, Floor[(Sqrt[8*n + 1] - 1)/2]}], {n, 2, 60}]}] (* Vaclav Kotesovec, Sep 26 2016 *)
ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];
Table[Length[Select[IntegerPartitions[n],ck[#]==0&]],{n,0,30}] (* Gus Wiseman, Apr 02 2021 *)

[[p.crank() for p in Partitions(n)].count(0) for n in (1..20)] # Peter Luschny, Sep 15 2014

a(n) = A000041(n) - 2*A001522(n). a(n) = A064391(n, 0).
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi / (3 * 2^(9/2) * n^(3/2)). - Vaclav Kotesovec, May 06 2018
a(n > 1) = A064428(n) - A001522(n), where A001522/A064428 count odd/even-length compositions with alternating parts strictly decreasing. - Gus Wiseman, Apr 02 2021
From Peter Bala, Feb 03 2024: (Start)
For n >= 2, a(n) = A188674(n+1) - A188674(n) (Hopkins and Sellers, Proposition 7).
Equivalently, the g.f. A(x) = (1 - x) * Sum_{n >= 1} x^(n*(n+2)) / Product{k = 1..n} (1 - x^k)^2. (End)

More terms from Reiner Martin, Dec 26 2001

A338470 Number of integer partitions of n with no part dividing all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 13, 7, 23, 21, 33, 35, 65, 55, 104, 97, 151, 166, 252, 235, 377, 399, 549, 591, 846, 858, 1237, 1311, 1749, 1934, 2556, 2705, 3659, 3991, 5090, 5608, 7244, 7841, 10086, 11075, 13794, 15420, 19195, 21003, 26240, 29089, 35483

Offset: 0

Gus Wiseman, Mar 23 2021

nonn

Alternative name: Number of integer partitions of n that are empty or have smallest part not dividing all the others.

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (32)  .  (43)   (53)   (54)    (64)    (65)     (75)
           (52)   (332)  (72)    (73)    (74)     (543)
           (322)         (432)   (433)   (83)     (552)
                         (522)   (532)   (92)     (732)
                         (3222)  (3322)  (443)    (4332)
                                         (533)    (5322)
                                         (542)    (33222)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

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

The complement is A083710 (strict: A097986).
The strict case is A341450.
The Heinz numbers of these partitions are A342193.
The dual version is A343341.
The case with maximum part not divisible by all the others is A343342.
The case with maximum part divisible by all the others is A343344.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A001787 count normal multisets with a selected position.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
A276024 counts positive subset sums.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001792, A064391, A064410, A066186, A067824, A083711, A098965, A264401, A339562, A339563.

Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]
(* Second program: *)
a[n_] := If[n == 0, 1, PartitionsP[n] - Sum[PartitionsP[d-1], {d, Divisors[n]}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 09 2021, after Andrew Howroyd *)

a(n)={numbpart(n) - if(n, sumdiv(n, d, numbpart(d-1)))} \\ Andrew Howroyd, Mar 25 2021

a(n) = A000041(n) - Sum_{d|n} A000041(d-1) for n > 0. - Andrew Howroyd, Mar 25 2021

A342192 Heinz numbers of partitions of crank 0.

Original entry on oeis.org

6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 100, 106, 118, 122, 134, 140, 142, 146, 158, 166, 178, 194, 196, 202, 206, 214, 218, 220, 226, 254, 260, 262, 274, 278, 298, 300, 302, 308, 314, 326, 334, 340, 346, 358, 362, 364, 380, 382, 386, 394, 398

Offset: 1

Gus Wiseman, Apr 05 2021

nonn

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.

See A257989 or the program for a definition of crank of a partition.

			The sequence of terms together with their prime indices begins:
      6: {1,2}        106: {1,16}       218: {1,29}
     10: {1,3}        118: {1,17}       220: {1,1,3,5}
     14: {1,4}        122: {1,18}       226: {1,30}
     22: {1,5}        134: {1,19}       254: {1,31}
     26: {1,6}        140: {1,1,3,4}    260: {1,1,3,6}
     34: {1,7}        142: {1,20}       262: {1,32}
     38: {1,8}        146: {1,21}       274: {1,33}
     46: {1,9}        158: {1,22}       278: {1,34}
     58: {1,10}       166: {1,23}       298: {1,35}
     62: {1,11}       178: {1,24}       300: {1,1,2,3,3}
     74: {1,12}       194: {1,25}       302: {1,36}
     82: {1,13}       196: {1,1,4,4}    308: {1,1,4,5}
     86: {1,14}       202: {1,26}       314: {1,37}
     94: {1,15}       206: {1,27}       326: {1,38}
    100: {1,1,3,3}    214: {1,28}       334: {1,39}

Indices of zeros in A257989.
A000005 counts constant partitions.
A000041 counts partitions (strict: A000009).
A001522 counts partitions of positive crank.
A003242 counts anti-run compositions.
A064391 counts partitions by crank.
A064428 counts partitions of nonnegative crank.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
Cf. A000726, A008965, A056239, A112798, A124010, A130091, A325351, A325352.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];
Select[Range[100],ck[primeMS[#]]==0&]

A115995 Sum of the sizes of the Durfee squares of all partitions of n.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 16, 23, 36, 52, 76, 106, 152, 207, 286, 386, 522, 691, 920, 1202, 1576, 2038, 2636, 3373, 4320, 5478, 6944, 8738, 10984, 13717, 17116, 21232, 26308, 32441, 39944, 48977, 59970, 73147, 89090, 108151, 131090, 158417, 191166, 230049, 276444

Offset: 0

Emeric Deutsch, Feb 11 2006

nonn

Also sum of positive cranks of all partitions of n, n>1; see A064391. - Vladeta Jovovic, Oct 20 2006

This sequence, its author and the author of the above comment were mentioned in the Andrews-Chan-Kim paper, where it is called C_1 (see the remark on page 6). - Omar E. Pol, Apr 06 2012

			a(4) = 6 because the partitions [4], [3,1], [2,2], [2,1,1] and [1,1,1,1] of 4 have Durfee squares of sizes 1,1,2,1 and 1, respectively.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Alois P. Heinz, Table of n, a(n) for n = 0..3000
George E. Andrews, Partitions and Durfee Dissection
George E. Andrews, Song Heng Chan, and Byungchan Kim, The odd moments of ranks and cranks
George E. Andrews, Frank G. Garvan, and Jie Liang, Self-conjugate vector partitions and the parity of the spt-function.
Atul Dixit, Bibekananda Maji, Partition implications of a new three parameter q-series identity, arXiv:1806.04424 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Durfee Square.

Cf. A115994, A115720, A115721, A115722.

g:= add(k*z^(k^2)/mul((1-z^j)^2,j=1..k),k=1..10): gser:=series(g,z=0,56): seq(coeff(gser,z,n), n=0..52);
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(add(b(k, d)*b(n-d^2-k, d), k=0..n-d^2)*d, d=1..isqrt(n)):
seq(a(n), n=0..70);  # Alois P. Heinz, Apr 09 2012
# Third Maple program, based on Theorem 1 of Andrews-Chan-Kim:
M:=101;
qinf:=mul(1-q^i,i=1..M);
qinf:=series(qinf,q,M);
C1:=add((-1)^(n+1)*q^(n*(n+1)/2)/(1-q^n),n=1..M);
C1:=series(C1/qinf,q,M);
seriestolist(%); # N. J. A. Sloane, Sep 04 2012

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]] ; a[n_] := Sum[ Sum[b[k, d]*b[n - d^2 - k, d], {k, 0, n - d^2}]*d, {d, 1, Sqrt[n]}]; Table [a[n], {n, 0, 70}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)

N=66; x='x+O('x^N); concat([0], Vec( sum(n=0,N, n*x^(n^2) / prod(k=1,n, 1-x^k)^2))) \\ Joerg Arndt, Mar 26 2014

[sum(p.frobenius_rank() for p in Partitions(n)) for n in range(45)] # Peter Luschny, Sep 15 2014

G.f.: Sum_{k>=1} (k*z^(k^2) / Product_{j=1..k} (1 - z^j)^2 ).
a(n) = Sum_{k=1..floor(sqrt(n))} k*A115994(n,k).
Convolution of A067742 and A000041. - Vladeta Jovovic, Oct 20 2006
a(n) = A195012(n) + A209616(n), n >= 1. - Omar E. Pol, Apr 06 2012
a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Jan 02 2019

Edited and verified by Franklin T. Adams-Watters, Mar 11 2006

A339662 Greatest gap in the partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 2, 4, 0, 5, 3, 1, 0, 6, 0, 7, 2, 3, 4, 8, 0, 2, 5, 1, 3, 9, 0, 10, 0, 4, 6, 2, 0, 11, 7, 5, 2, 12, 3, 13, 4, 1, 8, 14, 0, 3, 2, 6, 5, 15, 0, 4, 3, 7, 9, 16, 0, 17, 10, 3, 0, 5, 4, 18, 6, 8, 2, 19, 0, 20, 11, 1, 7, 3, 5, 21, 2, 1, 12

Offset: 1

Gus Wiseman, Apr 20 2021

nonn

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.
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.
Also the index of the greatest prime, up to the greatest prime index of n, not dividing n. A prime index of n is a number m such that prime(m) divides n.

George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
FindStat, Dyson's crank of a partition.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Positions of first appearances are A000040.
Positions of 0's are A055932.
The version for positions of 1's in reversed binary expansion is A063250.
The prime itself (not just the index) is A079068.
The version for crank is A257989.
The minimal instead of maximal version is A257993.
The version for greatest difference is A286469 or A286470.
Positive integers by Heinz weight and image are counted by A339737.
Positions of 1's are A339886.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A005117, A018818, A029707, A064391, A098743, A264401, A325351, A333214, A342192.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[maxgap[primeMS[n]],{n,100}]

a(n) = A000720(A079068(n)).

A257989 The crank of the partition having Heinz number n.

Original entry on oeis.org

-1, 2, -2, 3, 0, 4, -3, 2, 0, 5, -2, 6, 0, 3, -4, 7, 1, 8, -1, 4, 0, 9, -3, 3, 0, 2, -1, 10, 1, 11, -5, 5, 0, 4, -2, 12, 0, 6, -3, 13, 1, 14, -1, 3, 0, 15, -4, 4, 1, 7, -1, 16, 2, 5, -2, 8, 0, 17, -1, 18, 0, 4, -6, 6, 1, 19, -1, 9, 1, 20, -3, 21, 0, 3, -1, 5, 1, 22, -4, 2, 0, 23, -1, 7, 0, 10, -2, 24, 2, 6, -1

Offset: 2

Emeric Deutsch, May 18 2015

sign

The crank of a partition p is defined to be (i) the largest part of p if there is no 1 in p and (ii) (the number of parts larger than the number of 1's) minus (the number of 1's).
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n, the subprogram b yields the number of 1's in the partition with Heinz number n and the subprogram c yields the number of parts that are larger than the number of 1's in the partition with the Heinz number n.

			a(12) = - 2 because the partition with Heinz number 12 = 2*2*3 is [1,1,2], the number of parts larger than the number of 1's is 0 and the number of 1's is 2; 0 - 2 = -2.
a(945) = 4 because the partition with Heinz number 945 = 3^3 * 5 * 7 is [2,2,2,3,4] which has no part 1; the largest part is 4.
From _Gus Wiseman_, Apr 05 2021: (Start)
The partitions (center) with each Heinz number (left), and the corresponding terms (right):
   2:    (1)    -> -1
   3:    (2)    ->  2
   4:   (1,1)   -> -2
   5:    (3)    ->  3
   6:   (2,1)   ->  0
   7:    (4)    ->  4
   8:  (1,1,1)  -> -3
   9:   (2,2)   ->  2
  10:   (3,1)   ->  0
  11:    (5)    ->  5
  12:  (2,1,1)  -> -2
  13:    (6)    ->  6
  14:   (4,1)   ->  0
  15:   (3,2)   ->  3
  16: (1,1,1,1) -> -4
(End)

Alois P. Heinz, Table of n, a(n) for n = 2..10000
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
B. C. Berndt, H. H. Chan, S. H. Chan, W.-C. Liaw, Cranks and dissections in Ramanujan's lost notebook, J. Comb. Theory, Ser. A, 109, 2005, 91-120.
B. C. Berndt, H. H. Chan, S. H. Chan, W.-C. Liaw, Cranks - really the final problem, Ramanujan J., 23, 2010, 3-15.
G. E. Andrews, K. Ono, Ramanujan's congruences and Dyson's crank, Proc. Natl. Acad. Sci. USA, 102, 2005, 15277.
FindStat, Dyson's crank of a partition.
K. Mahlburg, Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Natl. Acad. Sci. USA, 102, 2005, 15373-15376.
Wikipedia, Crank of a partition

Cf. A215366, A257988.
Indices of zeros are A342192.
A001522 counts partitions of crank 0.
A003242 counts anti-run compositions.
A064391 counts partitions by crank.
A064428 counts partitions of nonnegative crank.
Cf. A000005, A000726, A056239, A112798, A124010, A224958, A325351, A325352.

with(numtheory): a := proc (n) local B, b, c: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do; [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: b := proc (n) if `mod`(n, 2) = 1 then 0 else 1+b((1/2)*n) end if end proc: c := proc (n) local b, B, ct, i: b := proc (n) if `mod`(n, 2) = 1 then 0 else 1+b((1/2)*n) end if end proc: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: ct := 0: for i to bigomega(n) do if b(n) < B(n)[i] then ct := ct+1 else  end if end do: ct end proc: if b(n) = 0 then max(B(n)) else c(n)-b(n) end if end proc: seq(a(n), n = 2 .. 150);

B[n_] := Module[{nn, j, m}, nn =  FactorInteger[n]; For[j = 1, j <= Length[nn], j++, m[j] = nn[[j]]]; Flatten[Table[Table[PrimePi[m[i][[1]]], {q, 1, m[i][[2]]}], {i, 1, Length[nn]}]]];
b[n_] := b[n] = If[OddQ[n], 0, 1 + b[n/2]];
c[n_] := Module[{ct, i}, ct = 0; For[i = 1, i <= PrimeOmega[n], i++, If[ b[n] < B[n][[i]], ct++]]; ct];
a[n_] := If[b[n] == 0, Max[B[n]], c[n] - b[n]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Apr 25 2017, after Emeric Deutsch *)
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];
Table[ck[primeMS[n]],{n,2,30}] (* Gus Wiseman, Apr 05 2021 *)

A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 1, 2, 2, 1, 1, 0, 5, 1, 3, 2, 2, 1, 1, 0, 6, 2, 3, 4, 3, 2, 1, 1, 0, 8, 2, 4, 5, 4, 3, 2, 1, 1, 0, 10, 2, 5, 7, 6, 5, 3, 2, 1, 1, 0, 12, 3, 6, 8, 9, 6, 5, 3, 2, 1, 1, 0, 15, 3, 8, 11, 11, 10, 7, 5, 3, 2, 1, 1, 0

Offset: 0

Gus Wiseman, Apr 20 2021

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.

			Triangle begins:
   1
   1   0
   1   1   0
   2   0   1   0
   2   1   1   1   0
   3   1   1   1   1   0
   4   1   2   2   1   1   0
   5   1   3   2   2   1   1   0
   6   2   3   4   3   2   1   1   0
   8   2   4   5   4   3   2   1   1   0
  10   2   5   7   6   5   3   2   1   1   0
  12   3   6   8   9   6   5   3   2   1   1   0
  15   3   8  11  11  10   7   5   3   2   1   1   0
  18   4   9  13  15  13  10   7   5   3   2   1   1   0
  22   5  10  17  19  18  14  11   7   5   3   2   1   1   0
  27   5  13  20  24  23  20  14  11   7   5   3   2   1   1   0
For example, row n = 9 counts the following partitions:
  (3321)       (432)   (333)      (54)      (522)    (63)    (72)   (81)  (9)
  (22221)      (3222)  (4311)     (441)     (531)    (621)   (711)
  (32211)              (33111)    (4221)    (5211)   (6111)
  (222111)             (3111111)  (42111)   (51111)
  (321111)                        (411111)
  (2211111)
  (21111111)
  (111111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Column k = 0 is A000009.
Row sums are A000041.
Central diagonal is A000041.
Column k = 1 is A087897.
The version for least gap is A264401, with Heinz number encoding A257993.
The version for greatest difference is A286469 or A286470.
An encoding (of greatest gap) using Heinz numbers is A339662.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A048004 counts compositions by greatest part.
A056239 adds up prime indices, row sums of A112798.
A064391 is the version for crank.
A064428 counts partitions of nonnegative crank.
A073491 list numbers with gap-free prime indices.
A107428 counts gap-free compositions.
A238709/A238710 counts partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A002110, A018818, A063250, A088860, A098743, A279945.

maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[Length[Select[IntegerPartitions[n],maxgap[#]==k&]],{n,0,15},{k,0,n}]

S(n,k)={if(k>n, O(x*x^n), x^k*(S(n-k,k+1) + 1)/(1 - x^k))}
ColGf(k,n) = {(k==0) + S(n,k+1)/prod(j=1, k-1, 1 - x^j + O(x^max(1,n-k)))}
A(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(M=A(10)); for(i=1, #M, print(M[i,1..i])) } \\ Andrew Howroyd, Jan 13 2024

Offset corrected by Andrew Howroyd, Jan 13 2024

A342343 Number of strict compositions of n with alternating parts strictly decreasing.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 10, 13, 18, 27, 32, 44, 55, 73, 97, 121, 151, 194, 240, 299, 384, 465, 576, 706, 869, 1051, 1293, 1572, 1896, 2290, 2761, 3302, 3973, 4732, 5645, 6759, 7995, 9477, 11218, 13258, 15597, 18393, 21565, 25319, 29703, 34701, 40478, 47278, 54985

Offset: 0

Gus Wiseman, Apr 01 2021

nonn

These are finite odd-length sequences q of distinct positive integers summing to n such that q(i) > q(i+2) for all possible i.

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)    (2,6)
                          (3,2)  (4,2)    (3,4)    (3,5)
                          (4,1)  (5,1)    (4,3)    (5,3)
                                 (2,3,1)  (5,2)    (6,2)
                                 (3,1,2)  (6,1)    (7,1)
                                 (3,2,1)  (2,4,1)  (2,5,1)
                                          (4,1,2)  (3,4,1)
                                          (4,2,1)  (4,1,3)
                                                   (4,3,1)
                                                   (5,1,2)
                                                   (5,2,1)

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

The non-strict case is A000041 (see A342528 for a bijective proof).
The non-strict odd-length case is A001522.
Strict compositions in general are counted by A032020
The non-strict even-length case is A064428.
The case of reversed partitions is A065033.
A000726 counts partitions with alternating parts unequal.
A003242 counts anti-run compositions.
A027193 counts odd-length compositions.
A034008 counts even-length compositions.
A064391 counts partitions by crank.
A064410 counts partitions of crank 0.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
A325548 counts compositions with strictly decreasing differences.
A342194 counts strict compositions with equal differences.
A342527 counts compositions with alternating parts equal.
Cf. A000009, A000670, A008965, A062968, A065608, A109613, A114921, A325546, A332304, A332305, A342532.

ici[q_]:=And@@Table[q[[i]]>q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],ici]],{n,0,15}]

seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=0, n, binomial(k, k\2) * polcoef(p,k,y)))} \\ Andrew Howroyd, Apr 16 2021

G.f.: Sum_{k>=0} binomial(k,floor(k/2)) * [y^k](Product_{j>=1} 1 + y*x^j). - Andrew Howroyd, Apr 16 2021

cp's OEIS Frontend

A001522 Number of n-stacks with strictly receding walls, or the number of Type A partitions of n in the sense of Auluck (1951).

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A064428 Number of partitions of n with nonnegative crank.

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A064410 Number of partitions of n with zero crank.

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A338470 Number of integer partitions of n with no part dividing all the others.

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A342192 Heinz numbers of partitions of crank 0.

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A115995 Sum of the sizes of the Durfee squares of all partitions of n.

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