A325548 -id:A325548 - cp's OEIS frontend

A325545 Number of compositions of n with distinct differences.

1, 1, 2, 3, 7, 13, 17, 34, 59, 105, 166, 279, 442, 730, 1157, 1927, 3045, 4741, 7527, 11667, 18048, 27928, 43334, 65861, 101385, 153404, 232287, 347643, 523721, 780083, 1165331, 1725966, 2561625, 3773838, 5561577, 8151209, 11920717, 17364461, 25269939, 36635775

Offset: 0

Gus Wiseman, May 10 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)   (3)   (4)    (5)     (6)
       (11)  (12)  (13)   (14)    (15)
             (21)  (22)   (23)    (24)
                   (31)   (32)    (33)
                   (112)  (41)    (42)
                   (121)  (113)   (51)
                   (211)  (122)   (114)
                          (131)   (132)
                          (212)   (141)
                          (221)   (213)
                          (311)   (231)
                          (1121)  (312)
                          (1211)  (411)
                                  (1131)
                                  (1221)
                                  (1311)
                                  (2112)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..70
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A059966, A070211, A175342, A242882, A325325, A325352, A325546, A325547, A325548, A325551, A325554.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Differences[#]&]],{n,0,15}]

More terms from Alois P. Heinz, May 11 2019

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

A175342 Number of arithmetic progressions (where the difference between adjacent terms is either positive, 0, or negative) of positive integers that sum to n.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 8, 10, 15, 14, 12, 22, 14, 18, 28, 21, 18, 34, 20, 28, 38, 28, 24, 46, 31, 32, 48, 38, 30, 62, 32, 40, 58, 42, 46, 73, 38, 46, 68, 58, 42, 84, 44, 56, 90, 56, 48, 94, 55, 70, 90, 66, 54, 106, 70, 74, 100, 70, 60, 130, 62, 74, 118, 81, 82, 130, 68, 84, 120

Offset: 1

Leroy Quet, Apr 17 2010

nonn

			From _Gus Wiseman_, May 15 2019: (Start)
The a(1) = 1 through a(8) = 10 compositions with equal differences:
  (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)
                    (1111)  (41)     (42)      (43)       (44)
                            (11111)  (51)      (52)       (53)
                                     (123)     (61)       (62)
                                     (222)     (1111111)  (71)
                                     (321)                (2222)
                                     (111111)             (11111111)
(End)

Lars Blomberg, Table of n, a(n) for n = 1..10000
Lars Blomberg, C# program for calculating b-file.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000005, A000079, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A070211, A127938, A175327, A325328, A325407, A325545, A325546, A325547, A325548, A325557, A325558.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Differences[#]&]],{n,0,15}] (* returns a(0) = 1, Gus Wiseman, May 15 2019*)

a(n) = 2*A049988(n) - A000005(n).

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

Edited and extended by Max Alekseyev, May 03 2010

A325557 Number of compositions of n with equal differences up to sign.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 13, 12, 20, 24, 25, 29, 49, 40, 50, 64, 86, 80, 105, 102, 164, 175, 186, 208, 325, 316, 382, 476, 624, 660, 814, 961, 1331, 1500, 1739, 2140, 2877, 3274, 3939, 4901, 6345, 7448, 9054, 11157, 14315, 17181, 20769, 25843, 32947, 39639, 48257, 60075

Offset: 0

Gus Wiseman, May 11 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(8) = 20 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)    (123)     (61)       (62)
                            (11111)  (141)     (151)      (71)
                                     (222)     (232)      (161)
                                     (321)     (313)      (242)
                                     (1212)    (12121)    (323)
                                     (2121)    (1111111)  (1232)
                                     (111111)             (1313)
                                                          (2123)
                                                          (2222)
                                                          (2321)
                                                          (3131)
                                                          (3212)
                                                          (21212)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A047966, A049988, A070211, A098504, A173258, A175342, A325545, A325546, A325547, A325548, A325552, A325558.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Abs[Differences[#]]&]],{n,0,15}]

step(R,n,s)={matrix(n, n, i, j, if(i>j, if(j>s, R[i-j, j-s]) + if(j+s<=n, R[i-j, j+s])) )}
w(n,s)={my(R=matid(n), t=0); while(R, R=step(R,n,s); t+=vecsum(R[n,])); t}
a(n) = {numdiv(max(1,n)) + sum(s=1, n-1, w(n,s))} \\ Andrew Howroyd, Aug 22 2019

a(26)-a(42) from Lars Blomberg, May 30 2019

Terms a(43) and beyond from Andrew Howroyd, Aug 22 2019

A070211 Number of compositions (ordered partitions) of n that are concave-down sequences.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 14, 18, 24, 34, 42, 52, 68, 82, 101, 126, 147, 175, 213, 246, 289, 344, 392, 453, 530, 598, 687, 791, 885, 1007, 1151, 1276, 1438, 1629, 1806, 2018, 2262, 2490, 2775, 3091, 3387, 3754, 4165, 4542, 5011, 5527, 6012, 6600, 7245, 7864, 8614

Offset: 0

Pontus von Brömssen, May 07 2002

Here, a finite sequence is concave if each term (other than the first or last) is at least the average of the two adjacent terms. - Eric M. Schmidt, Sep 29 2013

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1). Then a(n) is the number of compositions of n with weakly decreasing differences. - Gus Wiseman, May 15 2019

			Out of the 8 ordered partitions of 4, only 2+1+1 and 1+1+2 are not concave, so a(4)=6.
From _Gus Wiseman_, May 15 2019: (Start)
The a(1) = 1 through a(6) = 14 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (1111)  (122)    (51)
                            (131)    (123)
                            (221)    (132)
                            (11111)  (141)
                                     (222)
                                     (231)
                                     (321)
                                     (1221)
                                     (111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..500
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A001523 (weakly unimodal compositions), A069916, A175342, A320466, A325361 (concave-down partitions), A325545, A325546 (concave-up compositions), A325547, A325548, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@Differences[#]&]],{n,0,15}] (* Gus Wiseman, May 15 2019 *)

def A070211(n) : return sum(all(2*p[i] >= p[i-1] + p[i+1] for i in range(1, len(p)-1)) for p in Compositions(n)) # Eric M. Schmidt, Sep 29 2013

Name edited by Gus Wiseman, May 15 2019

A325546 Number of compositions of n with weakly increasing differences.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 19, 28, 41, 62, 87, 120, 170, 228, 303, 408, 534, 689, 899, 1145, 1449, 1842, 2306, 2863, 3571, 4398, 5386, 6610, 8039, 9716, 11775, 14157, 16938, 20293, 24166, 28643, 33995, 40134, 47199, 55540, 65088, 75994, 88776, 103328, 119886, 139126

Offset: 0

Gus Wiseman, May 10 2019

nonn

Also compositions of n whose plot is concave-up.
A composition of n is a finite sequence of positive integers summing to n.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (112)   (41)     (42)
                    (211)   (113)    (51)
                    (1111)  (212)    (114)
                            (311)    (123)
                            (1112)   (213)
                            (2111)   (222)
                            (11111)  (312)
                                     (321)
                                     (411)
                                     (1113)
                                     (2112)
                                     (3111)
                                     (11112)
                                     (21111)
                                     (111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A007294, A008965, A070211 (concave-down compositions), A173258, A175342, A240026, A325360, A325545, A325547, A325548, A325552, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],LessEqual@@Differences[#]&]],{n,0,15}]

\\ Row sums of R(n) give A007294 (=breakdown by width).
R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L,v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-w-1)\t + 1, v[i-w-(k-1)*t]))); Mat(L)}
seq(n)={my(M=R(n)); Vec(1 + sum(i=1, n, my(p=sum(w=1, min(#M,n\i), x^(w*i)*sum(j=1, n-i*w, x^j*M[j,w])));  x^i/(1 - x^i)*(1 + p + O(x*x^(n-i)))^2))} \\ Andrew Howroyd, Aug 28 2019

More terms from Alois P. Heinz, May 11 2019

A325547 Number of compositions of n with strictly increasing differences.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 11, 18, 24, 30, 45, 57, 71, 96, 120, 148, 192, 235, 286, 354, 431, 518, 628, 752, 893, 1063, 1262, 1482, 1744, 2046, 2386, 2775, 3231, 3733, 4305, 4977, 5715, 6536, 7507, 8559, 9735, 11112, 12608, 14252, 16177, 18265, 20553, 23204, 26090, 29223

Offset: 0

Gus Wiseman, May 10 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

			The a(1) = 1 through a(6) = 11 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)
       (11)  (12)  (13)   (14)   (15)
             (21)  (22)   (23)   (24)
                   (31)   (32)   (33)
                   (112)  (41)   (42)
                   (211)  (113)  (51)
                          (212)  (114)
                          (311)  (213)
                                 (312)
                                 (411)
                                 (2112)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A034297, A070211, A175342, A179269, A179254, A240027, A325545, A325546, A325548, A325552, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Differences[#]&]],{n,0,15}]

\\ Row sums of R(n) give A179269 (breakdown by width)
R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L,v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-1)\t, v[i-k*t]))); Mat(L)}
seq(n)={my(M=R(n)); Vec(1 + sum(i=1, n, my(p=sum(w=1, min(#M,n\i), x^(w*i)*sum(j=1, n-i*w, x^j*M[j,w])));  x^i*(1 + x^i)*(1 + p + O(x*x^(n-i)))^2))} \\ Andrew Howroyd, Aug 27 2019

a(26)-a(42) from Lars Blomberg, May 30 2019

Terms a(43) and beyond from Andrew Howroyd, Aug 27 2019

A068322 Number of arithmetic progressions of positive odd integers, strictly increasing with sum n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 3, 3, 5, 1, 4, 1, 5, 4, 5, 1, 7, 2, 6, 5, 8, 1, 7, 1, 9, 6, 8, 2, 11, 1, 9, 7, 12, 1, 10, 1, 12, 10, 11, 1, 15, 2, 12, 9, 15, 1, 13, 3, 16, 10, 14, 1, 18, 1, 15, 12, 20, 4, 17, 1, 19, 12, 17, 1, 22, 1, 18, 16, 22, 2, 20, 1, 24, 15, 20, 1, 25, 5, 21, 15, 26

Offset: 1

Naohiro Nomoto, Feb 27 2002

			From _Petros Hadjicostas_, Sep 29 2019: (Start)
a(12) = 3 because we have the following arithmetic progressions of odd numbers, strictly increasing with sum n=12: 1+11, 3+9, and 5+7.
a(13) = 1 because we have only the following arithmetic progressions of odd numbers, strictly increasing with sum n=13: 13.
a(14) = 3 because we have the following arithmetic progressions of odd numbers, strictly increasing with sum n=14: 1+13, 3+11, and 5+9.
a(15) = 3 because we have the following arithmetic progressions of odd numbers, strictly increasing with sum n=15: 15, 3+5+7, and 1+5+9.
(End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A068323, A068324, A070211, A127938, A175327, A325328, A325407, A325545, A325546, A325547, A325548.

From Petros Hadjicostas, Oct 01 2019: (Start)
a(n) = A068324(n) - A001227(n) + (1/2) * (1 - (-1)^n).
G.f.: x/(1 - x^2) + Sum_{m >= 2} x^(m^2)/((1 - x^(2*m)) * (1 - x^(m*(m-1)))).
(End)

A068324 Number of nondecreasing arithmetic progressions of positive odd integers with sum n.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 3, 4, 4, 2, 5, 2, 5, 6, 6, 2, 7, 2, 7, 7, 7, 2, 9, 4, 8, 8, 10, 2, 11, 2, 10, 9, 10, 5, 14, 2, 11, 10, 14, 2, 14, 2, 14, 15, 13, 2, 17, 4, 15, 12, 17, 2, 17, 6, 18, 13, 16, 2, 22, 2, 17, 17, 21, 7, 21, 2, 21, 15, 21, 2, 25, 2, 20, 21, 24, 5, 24, 2, 26, 19, 22, 2, 29, 8

Offset: 1

Naohiro Nomoto, Feb 27 2002

			From _Petros Hadjicostas_, Sep 29 2019: (Start)
a(6) = 3 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=6: 1+5, 3+3, and 1+1+1+1+1+1.
a(7) = 2 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=7: 7 and 1+1+1+1+1+1+1.
a(8) = 3 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=8: 1+7, 3+5, and 1+1+1+1+1+1+1+1.
(End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A068322, A068323, A127938, A175327, A325328, A325407, A325545, A325546, A325547, A325548.

From Petros Hadjicostas, Oct 01 2019: (Start)
a(n) = A068322(n) + A001227(n) - (1/2) * (1 - (-1)^n).
G.f.: x/(1 - x^2) + Sum_{m >= 2} x^m/((1 - x^(2*m)) * (1 - x^(m*(m-1)))).
(End)

Extended and edited by John W. Layman, Mar 15 2002

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

A325545 Number of compositions of n with distinct differences.

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A064428 Number of partitions of n with nonnegative crank.

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A175342 Number of arithmetic progressions (where the difference between adjacent terms is either positive, 0, or negative) of positive integers that sum to n.

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A325557 Number of compositions of n with equal differences up to sign.

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A070211 Number of compositions (ordered partitions) of n that are concave-down sequences.

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A325546 Number of compositions of n with weakly increasing differences.

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A325547 Number of compositions of n with strictly increasing differences.

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