A114921 -id:A114921 - 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

A065608 Sum of divisors of n minus the number of divisors of n.

Original entry on oeis.org

0, 1, 2, 4, 4, 8, 6, 11, 10, 14, 10, 22, 12, 20, 20, 26, 16, 33, 18, 36, 28, 32, 22, 52, 28, 38, 36, 50, 28, 64, 30, 57, 44, 50, 44, 82, 36, 56, 52, 82, 40, 88, 42, 78, 72, 68, 46, 114, 54, 87, 68, 92, 52, 112, 68, 112, 76, 86, 58, 156, 60, 92, 98, 120, 80, 136, 66, 120, 92

Offset: 1

Jason Earls, Nov 06 2001

Number of permutations p of {1,2,...,n} such that p(k)-k takes exactly two distinct values. Example: a(4)=4 because we have 4123, 3412, 2143 and 2341. - Max Alekseyev and Emeric Deutsch, Dec 22 2006
Number of solutions to the Diophantine equation xy + yz = n, with x,y,z >= 1.
In other words, number of ways to write n = (a + b) * k for positive integers a, b, k. - Gus Wiseman, Mar 25 2021
Not the same as A184396(n): a(66) = 136 while A184396(66) = 137. - Wesley Ivan Hurt, Dec 26 2013
From Gus Wiseman, Mar 25 2021: (Start)
Also the number of compositions of n into an even number of parts with alternating parts equal. These are finite even-length sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i. For example, the a(2) = 1 through a(8) = 11 compositions are:
  (1,1)  (1,2)  (1,3)      (1,4)  (1,5)          (1,6)  (1,7)
         (2,1)  (2,2)      (2,3)  (2,4)          (2,5)  (2,6)
                (3,1)      (3,2)  (3,3)          (3,4)  (3,5)
                (1,1,1,1)  (4,1)  (4,2)          (4,3)  (4,4)
                                  (5,1)          (5,2)  (5,3)
                                  (1,2,1,2)      (6,1)  (6,2)
                                  (2,1,2,1)             (7,1)
                                  (1,1,1,1,1,1)         (1,3,1,3)
                                                        (2,2,2,2)
                                                        (3,1,3,1)
                                                        (1,1,1,1,1,1,1,1)
The odd-length version is A062968.
The version with alternating parts weakly decreasing is A114921, or A342528 if odd-length compositions are included.
The version with alternating parts unequal is A342532, or A224958 if odd-length compositions are included (unordered: A339404/A000726).
Allowing odd lengths as well as even gives A342527.
(End)
Inverse Möbius transform of n-1. - Wesley Ivan Hurt, Jun 29 2024

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
M. Alekseyev, E. Deutsch, and J. H. Steelman, Solution to problem 11281, Amer. Math. Monthly, 116, No. 5, 2009, p. 465.
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_2(n).
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Masato Kobayashi, New recurrences for divisor sum functions and triangular numbers, arXiv:2207.05831 [math.NT], 2022.

Cf. A000203, A000005, A134857.
Starting (1, 2, 4, 4, 8, 6, ...), = row sums of triangle A077478. - Gary W. Adamson, Nov 12 2007
Starting with "1" = row sums of triangle A176919. - Gary W. Adamson, Apr 29 2010
Column k=2 of A125182.
A175342/A325545 count compositions with constant/distinct differences.
Cf. A001522, A002843, A008965, A010054, A064410, A064428, A325557, A342495.

List([1..100],n->Sigma(n)-Tau(n)); # Muniru A Asiru, Mar 19 2018

with(numtheory): seq(sigma(n)-tau(n),n=1..70); # Emeric Deutsch, Dec 22 2006

Table[DivisorSigma[1,n]-DivisorSigma[0,n], {n,100}] (* Wesley Ivan Hurt, Dec 26 2013 *)

a(n) = sigma(n) - numdiv(n); \\ Harry J. Smith, Oct 23 2009

from math import prod
from sympy import factorint
def A065608(n):
    f = factorint(n).items()
    return prod((p**(e+1)-1)//(p-1) for p, e in f)-prod(e+1 for p,e in f) # Chai Wah Wu, Jul 16 2022

a(n) = sigma(n) - d(n) = A000203(n) - A000005(n).
a(n) = Sum_{d|n} (d-1). - Wesley Ivan Hurt, Dec 26 2013
G.f.: Sum_{k>=1} x^(2*k)/(1-x^k)^2. - Benoit Cloitre, Apr 21 2003
G.f.: Sum_{n>=1} (n-1)*x^n/(1-x^n). - Joerg Arndt, Jan 30 2011
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(1-1/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 18 2018
G.f.: Sum_{n >= 1} q^(n^2)*( (n - 1) + q^n - (n - 1)*q^(2*n) )/(1 - q^n)^2 - differentiate equation 1 in Arndt with respect to t, then set x = q and t = q. - Peter Bala, Jan 22 2021
a(n) = A342527(n) - A062968(n). - Gus Wiseman, Mar 25 2021
a(n) = n * A010054(n) - Sum_{k>=1} a(n - k*(k+1)/2), assuming a(n) = 0 for n <= 0 (Kobayashi, 2022). - Amiram Eldar, Jun 23 2023

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

A342528 Number of compositions with alternating parts weakly decreasing (or weakly increasing).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 20, 32, 51, 79, 121, 182, 272, 399, 582, 839, 1200, 1700, 2394, 3342, 4640, 6397, 8771, 11955, 16217, 21878, 29386, 39285, 52301, 69334, 91570, 120465, 157929, 206313, 268644, 348674, 451185, 582074, 748830, 960676, 1229208, 1568716, 1997064

Offset: 0

Gus Wiseman, Mar 24 2021

nonn

These are finite sequences q of positive integers summing to n such that q(i) >= q(i+2) for all possible i.
The strict case (alternating parts are strictly decreasing) is A000041. Is there a bijective proof?
Yes. Construct a Ferrers diagram by placing odd parts horizontally and even parts vertically in a fishbone pattern. The resulting Ferrers diagram will be for an ordinary partition and the process is reversible. It does not appear that this method can be applied to give a formula for this sequence. - Andrew Howroyd, Mar 25 2021

			The a(1) = 1 through a(6) = 20 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (211)   (131)    (51)
                    (1111)  (212)    (141)
                            (221)    (222)
                            (311)    (231)
                            (1211)   (312)
                            (2111)   (321)
                            (11111)  (411)
                                     (1212)
                                     (1311)
                                     (2121)
                                     (2211)
                                     (3111)
                                     (12111)
                                     (21111)
                                     (111111)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from Andrew Howroyd)
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The even-length case is A114921.
The version with alternating parts unequal is A224958 (unordered: A000726).
The version with alternating parts equal is A342527.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000203 adds up divisors.
A002843 counts compositions with all adjacent parts x <= 2y.
A003242 counts anti-run compositions.
A069916/A342492 = decreasing/increasing first quotients.
A070211/A325546 = weakly decreasing/increasing differences.
A175342/A325545 = constant/distinct differences.
A342495 = constant first quotients (unordered: A342496, strict: A342515, ranking: A342522).
Cf. A001522, A008965, A048004, A059966, A062968, A064410, A064428, A065608, A167606, A325557, A342519.

b:= proc(n, i, j) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, j)+b(n-i, min(n-i, j), min(n-i, i))))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..42);  # Alois P. Heinz, Jan 16 2025

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@Plus@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]

seq(n)={my(p=1/prod(k=1, n, 1-y*x^k + O(x*x^n))); Vec(1+sum(k=1, n, polcoef(p,k,y)*(polcoef(p,k-1,y) + polcoef(p,k,y))))} \\ Andrew Howroyd, Mar 24 2021

G.f.: Sum_{k>=0} ([y^k] P(x,y))*([y^k] (1 + y)*P(x,y)), where P(x,y) = Product_{k>=1} 1/(1 - y*x^k). - Andrew Howroyd, Jan 16 2025

Terms a(21) and beyond from Andrew Howroyd, Mar 24 2021

A342527 Number of compositions of n with alternating parts equal.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 11, 12, 16, 17, 21, 20, 29, 24, 31, 32, 38, 32, 46, 36, 51, 46, 51, 44, 69, 51, 61, 60, 73, 56, 87, 60, 84, 74, 81, 76, 110, 72, 91, 88, 115, 80, 123, 84, 117, 112, 111, 92, 153, 101, 132, 116, 139, 104, 159, 120, 161, 130, 141, 116, 205, 120, 151, 156, 178, 142, 195, 132, 183, 158

Offset: 0

Gus Wiseman, Mar 24 2021

nonn

These are finite sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i.

			The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (121)   (41)     (42)      (43)       (44)
                    (1111)  (131)    (51)      (52)       (53)
                            (212)    (141)     (61)       (62)
                            (11111)  (222)     (151)      (71)
                                     (1212)    (232)      (161)
                                     (2121)    (313)      (242)
                                     (111111)  (12121)    (323)
                                               (1111111)  (1313)
                                                          (2222)
                                                          (3131)
                                                          (21212)
                                                          (11111111)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The odd-length case is A062968.
The even-length case is A065608.
The version with alternating parts unequal is A224958 (unordered: A000726).
The version with alternating parts weakly decreasing is A342528.
A000005 counts constant compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000203 adds up divisors.
A002843 counts compositions with all adjacent parts x <= 2y.
A003242 counts anti-run compositions.
A175342 counts compositions with constant differences.
A342495 counts compositions with constant first quotients.
A342496 counts partitions with constant first quotients (strict: A342515, ranking: A342522).
Cf. A001522, A008965, A048004, A059966, A064410, A064428, A069916, A114921, A167606, A325545, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Plus@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]

a(n) = 1 + n + A000203(n) - 2*A000005(n).

a(n) = A065608(n) + A062968(n).

A238779 Number of palindromic partitions of n with greatest part of multiplicity 2.

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 2, 4, 3, 7, 6, 11, 9, 18, 15, 27, 23, 40, 35, 59, 51, 85, 75, 119, 106, 168, 150, 231, 208, 316, 286, 428, 388, 575, 525, 764, 700, 1012, 929, 1327, 1223, 1732, 1601, 2246, 2080, 2898, 2692, 3715, 3459, 4748, 4428, 6032, 5638, 7635, 7150

Offset: 1

Clark Kimberling, Mar 05 2014

Palindromic partitions are defined at A025065.

			a(8) counts these partitions (each written as a palindrome):  44, 323, 1331, 112211.

Cf. A025065, A087897, A238780, A114921.

z = 40; p[n_, k_] := Select[IntegerPartitions[n], (Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1) && (Count[#, Max[#]] == k) &]
Table[p[n, 1], {n, 1, 12}]
t1 = Table[Length[p[n, 1]], {n, 1, z}] (* A000009(n-1), n>=1 *)
Table[p[n, 2], {n, 1, 12}]
t2 = Table[Length[p[n, 2]], {n, 1, z}] (* A238779 *)
Table[p[n, 3], {n, 1, 12}]
t3 = Table[Length[p[n, 3]], {n, 1, z}] (* A087897(n-3), n>=3 *)
Table[p[n, 4], {n, 1, 12}]
t4 = Table[Length[p[n, 4]], {n, 1, z}] (* A238780 *)
(* Peter J. C. Moses, Mar 03 2014 *)

A247255 Triangular array read by rows: T(n,k) is the number of weakly unimodal partitions of n in which the greatest part occurs exactly k times, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 6, 1, 0, 1, 12, 2, 0, 0, 1, 21, 4, 1, 0, 0, 1, 38, 6, 2, 0, 0, 0, 1, 63, 11, 3, 1, 0, 0, 0, 1, 106, 16, 5, 2, 0, 0, 0, 0, 1, 170, 27, 7, 3, 1, 0, 0, 0, 0, 1, 272, 40, 11, 4, 2, 0, 0, 0, 0, 0, 1, 422, 63, 16, 6, 3, 1, 0, 0, 0, 0, 0, 1, 653, 92, 24, 8, 4, 2, 0, 0, 0, 0, 0, 0, 1, 986, 141, 34, 12, 5, 3, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Geoffrey Critzer, Nov 29 2014

These are called stack polyominoes in the Flajolet and Sedgewick reference.

			    1;
    1,  1;
    3,  0, 1;
    6,  1, 0, 1;
   12,  2, 0, 0, 1;
   21,  4, 1, 0, 0, 1;
   38,  6, 2, 0, 0, 0, 1;
   63, 11, 3, 1, 0, 0, 0, 1;
  106, 16, 5, 2, 0, 0, 0, 0, 1;
  170, 27, 7, 3, 1, 0, 0, 0, 0, 1;

P. Flajolet and R Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 46.

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=1-10 give: A006330, A114921, A226541, A320315, A320316, A320317, A320318, A320319, A320320, A320321.
Row sums give A001523.
Main diagonal gives A000012.

b:= proc(n, i) option remember; local r; expand(
      `if`(i>n, 0, `if`(irem(n, i, 'r')=0, x^r, 0)+
      add(b(n-i*j, i+1)*(j+1), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):
seq(T(n), n=1..14);  # Alois P. Heinz, Nov 29 2014

nn = 14; Table[
  Take[Drop[
     CoefficientList[
      Series[ Sum[
        u z^k/(1 - u z^k) Product[1/(1 - z^i), {i, 1, k - 1}]^2, {k,
         1, nn}], {z, 0, nn}], {z, u}], 1], n, {2, n + 1}][[n]], {n,
   1, nn}] // Grid

G.f.: Sum_{k>=1} y*x^k/(1 - y*x^k)/(Product_{i=1..k-1} (1 - x^i))^2.

For fixed k>=1, T(n,k) ~ Pi^(k-1) * (k-1)! * exp(2*Pi*sqrt(n/3)) / (2^(k+2) * 3^(k/2 + 1/4) * n^(k/2 + 3/4)). - Vaclav Kotesovec, Oct 24 2018

A342532 Number of even-length compositions of n with alternating parts distinct.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 9, 14, 28, 44, 83, 136, 250, 424, 757, 1310, 2313, 4018, 7081, 12314, 21650, 37786, 66264, 115802, 202950, 354858, 621525, 1087252, 1903668, 3330882, 5831192, 10204250, 17862232, 31260222, 54716913, 95762576, 167614445, 293356422, 513456686

Offset: 0

Gus Wiseman, Mar 28 2021

nonn

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

			The a(2) = 1 through a(7) = 14 compositions:
  (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)
                              (1,1,2,2)  (6,1)
                              (1,2,2,1)  (1,1,2,3)
                              (2,1,1,2)  (1,1,3,2)
                              (2,2,1,1)  (1,2,3,1)
                                         (1,3,2,1)
                                         (2,1,1,3)
                                         (2,3,1,1)
                                         (3,1,1,2)
                                         (3,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..500

The strictly decreasing version appears to be A064428 (odd-length: A001522).
The equal version is A065608 (A342527 with odds).
The weakly decreasing version is A114921 (A342528 with odds).
Including odds gives A224958.
A000726 counts partitions with alternating parts unequal.
A325545 counts compositions with distinct first differences.
A342529 counts compositions with distinct first quotients.
Cf. A000009, A000041, A003242, A008965, A032020, A059966, A062968, A064410, A070211, A106351, A175342, A325546.

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

\\ here gf gives A106351 as g.f.
gf(n, y)={1/(1 - sum(k=1, n, (-1)^(k+1)*x^k*y^k/(1-x^k) + O(x*x^n)))}
seq(n)={my(p=gf(n,y)); Vec(sum(k=0, n\2, polcoef(p,k,y)^2))} \\ Andrew Howroyd, Apr 16 2021

G.f.: 1 + Sum_{k>=1} B_k(x)^2 where B_k(x) is the g.f. of column k of A106351. - Andrew Howroyd, Apr 16 2021

Terms a(24) and beyond from Andrew Howroyd, Apr 16 2021

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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A064428 Number of partitions of n with nonnegative crank.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A065608 Sum of divisors of n minus the number of divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Python

Formula

A064410 Number of partitions of n with zero crank.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A342528 Number of compositions with alternating parts weakly decreasing (or weakly increasing).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A342527 Number of compositions of n with alternating parts equal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs