A070211 -id:A070211 - 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

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

A069916 Number of log-concave compositions (ordered partitions) of n.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 14, 20, 26, 36, 47, 60, 80, 102, 127, 159, 194, 236, 291, 355, 425, 514, 611, 718, 856, 1009, 1182, 1381, 1605, 1861, 2156, 2496, 2873, 3299, 3778, 4301, 4902, 5574, 6325, 7176, 8116, 9152, 10317, 11610, 13028, 14611, 16354, 18259, 20365

Offset: 0

Pontus von Brömssen, Apr 24 2002

These are compositions with weakly decreasing first quotients, where the first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3). - Gus Wiseman, Mar 16 2021

			Out of the 8 compositions of 4, only 2+1+1 and 1+1+2 are not log-concave, so a(4)=6.
From _Gus Wiseman_, Mar 15 2021: (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..300
Sean A. Irvine, Java program (github)
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A070211.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A001055 counts factorizations.
A002843 counts compositions with adjacent parts x <= 2y.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A003242 counts anti-run compositions.
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors summing to n.
Cf. A008965, A048004, A059966, A167606, A175342, A325547.

(* This program is not suitable for computing a large number of terms *)
compos[n_] := Permutations /@ IntegerPartitions[n] // Flatten[#, 1]&;
logConcaveQ[p_] := And @@ Table[p[[i]]^2 >= p[[i-1]]*p[[i+1]], {i, 2, Length[p]-1}]; a[n_] := Count[compos[n], p_?logConcaveQ]; Table[an = a[n]; Print["a(", n, ") = ", an]; a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 29 2016 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}] (* Gus Wiseman, Mar 15 2021 *)

def A069916(n) : return sum(all(p[i]^2 >= 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

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

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

A114921 Number of unimodal compositions of n+2 where the maximal part appears exactly twice.

Original entry on oeis.org

1, 0, 1, 2, 4, 6, 11, 16, 27, 40, 63, 92, 141, 202, 299, 426, 614, 862, 1222, 1694, 2362, 3242, 4456, 6054, 8229, 11072, 14891, 19872, 26477, 35050, 46320, 60866, 79827, 104194, 135703, 176008, 227791, 293702, 377874, 484554, 620011, 790952, 1006924

Offset: 0

Michael Somos, Jan 07 2006

nonn

Old name was: Expansion of a q-series.
a(n) is also the number of 2-colored partitions of n with the same number of parts in each color. - Shishuo Fu, May 30 2017
From Gus Wiseman, Mar 25 2021: (Start)
Also the number of even-length compositions of n with alternating parts weakly decreasing. Allowing odd lengths also gives A342528. The version with alternating parts strictly decreasing appears to be A064428. The a(2) = 1 through a(7) = 16 compositions 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)
                (1,1,1,1)  (4,1)      (4,2)          (4,3)
                           (1,2,1,1)  (5,1)          (5,2)
                           (2,1,1,1)  (1,2,1,2)      (6,1)
                                      (1,3,1,1)      (1,3,1,2)
                                      (2,1,2,1)      (1,4,1,1)
                                      (2,2,1,1)      (2,2,1,2)
                                      (3,1,1,1)      (2,2,2,1)
                                      (1,1,1,1,1,1)  (2,3,1,1)
                                                     (3,1,2,1)
                                                     (3,2,1,1)
                                                     (4,1,1,1)
                                                     (1,2,1,1,1,1)
                                                     (2,1,1,1,1,1)
(End)

			From _Joerg Arndt_, Jun 10 2013: (Start)
There are a(7)=16 such compositions of 7+2=9 where the maximal part appears twice:
  01:  [ 1 1 1 1 1 2 2 ]
  02:  [ 1 1 1 1 2 2 1 ]
  03:  [ 1 1 1 2 2 1 1 ]
  04:  [ 1 1 1 3 3 ]
  05:  [ 1 1 2 2 1 1 1 ]
  06:  [ 1 1 3 3 1 ]
  07:  [ 1 2 2 1 1 1 1 ]
  08:  [ 1 2 3 3 ]
  09:  [ 1 3 3 1 1 ]
  10:  [ 1 3 3 2 ]
  11:  [ 1 4 4 ]
  12:  [ 2 2 1 1 1 1 1 ]
  13:  [ 2 3 3 1 ]
  14:  [ 3 3 1 1 1 ]
  15:  [ 3 3 2 1 ]
  16:  [ 4 4 1 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
S. Fu and D. Tang, On a generalized crank for k-colored partitions, arXiv:1705.10067 [math.CO], 2017.
B. Kim and J. Lovejoy, Ramanujan-type partial theta identities and rank differences for special unimodal sequences, Annals of Combinatorics, 19 (2015), 705-733.

Cf. A226541 (max part appears three times), A188674 (max part m appears m times), A001523 (max part appears any number of times).
Column k=2 of A247255.
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.
A034008 counts even-length compositions.
A065608 counts even-length compositions with alternating parts equal.
A342528 counts compositions with alternating parts weakly decreasing.
A342532 counts even-length compositions with alternating parts unequal.
Cf. A000726, A001522, A008965, A062968, A064410, A064428, A069916, A070211, A175342, A224958, A342495, A342527.

max = 50; s = (1+Sum[2*(-1)^k*q^(k(k+1)/2), {k, 1, max}])/QPochhammer[q]^2+ O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, from 1st g.f. *)
wdw[q_]:=And@@Table[q[[i]]>=q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],EvenQ[Length[#]]&],wdw]],{n,0,15}] (* Gus Wiseman, Mar 25 2021 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, n\2, x^(2*k) / prod(i=1, k, 1 - x^i, 1 + x * O(x^n))^2), n))};

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

G.f.: 1 + Sum_{k>0} (x^k / ((1-x)(1-x^2)...(1-x^k)))^2 = (1 + Sum_{k>0} 2 (-1)^k x^((k^2+k)/2) ) / (Product_{k>0} (1 - x^k))^2.
G.f.: 1 + x*(1 - G(0))/(1-x) where G(k) = 1 - x/(1-x^(k+1))^2/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) = A006330(n) - A001523(n). - Vaclav Kotesovec, Jun 22 2015
a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (16 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Oct 24 2018

New name from Joerg Arndt, Jun 10 2013

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

A328220 Number of strict integer partitions of n with no pair of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 3, 10, 3, 11, 7, 12, 3, 19, 5, 18, 12, 23, 9, 36, 11, 33, 21, 40, 20, 58, 19, 58, 35, 70, 31, 98, 36, 101, 65, 112, 56, 155, 64, 164, 97, 188, 88, 250, 112, 256, 157, 293, 145, 392, 163, 399, 241, 461, 242

Offset: 0

Gus Wiseman, Oct 14 2019

nonn

			The a(2) = 1 through a(20) = 11 partitions (A..K = 10..20):
  2  3  4  5  6   7  8   9   A   B  C    D  E    F   G    H    I    J    K
              42     62  63  64     84      86   96  A6   863  A8   964  C8
                             82     93      A4   A5  C4   962  C6   A63  E6
                                    A2      C2   C3  E2        E4        F5
                                    642     842      862       F3        G4
                                                     A42       G2        I2
                                                               864       A64
                                                               963       A82
                                                               A62       C62
                                                               C42       E42
                                                                         8642

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

The non-strict case is A328187.
Partitions with all consecutive parts relatively prime are A328172, with strict case A328188.
Strict partitions with relatively prime parts are A078374.
Partitions with no consecutive divisibilities are A328171.
Cf. A000837, A018783, A070211, A289509, A318980, A325545, A328170, A328221, A328335, A328336.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]==1]&]],{n,0,30}]

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

A325548 Number of compositions of n with strictly decreasing differences.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 10, 13, 19, 23, 29, 38, 46, 55, 69, 80, 96, 115, 132, 154, 183, 207, 238, 276, 314, 356, 405, 455, 513, 579, 647, 724, 809, 897, 998, 1107, 1225, 1350, 1486, 1639, 1805, 1973, 2166, 2374, 2586, 2824, 3084, 3346, 3646, 3964, 4286, 4655, 5047

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(8) = 19 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)    (8)
       (11)  (12)  (13)   (14)   (15)    (16)   (17)
             (21)  (22)   (23)   (24)    (25)   (26)
                   (31)   (32)   (33)    (34)   (35)
                   (121)  (41)   (42)    (43)   (44)
                          (122)  (51)    (52)   (53)
                          (131)  (132)   (61)   (62)
                          (221)  (141)   (133)  (71)
                                 (231)   (142)  (134)
                                 (1221)  (151)  (143)
                                         (232)  (152)
                                         (241)  (161)
                                         (331)  (233)
                                                (242)
                                                (251)
                                                (332)
                                                (341)
                                                (431)
                                                (1331)

Alois P. Heinz, Table of n, a(n) for n = 0..400
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A011782, A000740, A008965, A070211, A175342, A179254, A320470, A325457, A325545, A325546, A325547, A325552, A325557.

b:= proc(n, l, d) option remember; `if`(n=0, 1, add(`if`(l=0 or
       j-l b(n, 0$2):
seq(a(n), n=0..52);  # Alois P. Heinz, Jan 27 2024

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

a(26)-a(44) from Lars Blomberg, May 30 2019

a(45)-a(52) from Alois P. Heinz, Jan 27 2024

cp's OEIS Frontend

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

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

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A114921 Number of unimodal compositions of n+2 where the maximal part appears exactly twice.

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A325546 Number of compositions of n with weakly increasing differences.

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