A336128 -id:A336128 - cp's OEIS frontend

A133494 Diagonal of the array of iterated differences of A047848.

1, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961, 68630377364883

Offset: 0

Paul Barry, Paul Curtz, Dec 23 2007

a(n) is the number of ways to choose a composition C, and then choose a composition of each part of C. - Geoffrey Critzer, Mar 19 2012
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the reptend length of 1/3^(n+1) in decimal. - Jianing Song, Nov 14 2018
Also the number of pairs of integer compositions, the first summing to n and the second with sum equal to the length of the first. If an integer composition is regarded as an arrow from sum to length, these are composable pairs, and the obvious composition operation founds a category of integer compositions. For example, we have (2,1,1,4) . (1,2,1) . (1,2) = (2,6), where dots represent the composition operation. The version without empty compositions is A000244. Composable triples are counted by 1 followed by A000302. The unordered version is A022811. - Gus Wiseman, Jul 14 2022

			From _Gus Wiseman_, Jul 15 2020: (Start)
The a(0) = 1 through a(3) = 9 ways to choose a composition of each part of a composition:
  ()  (1)  (2)      (3)
           (1,1)    (1,2)
           (1),(1)  (2,1)
                    (1,1,1)
                    (1),(2)
                    (2),(1)
                    (1),(1,1)
                    (1,1),(1)
                    (1),(1),(1)
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Index entries for linear recurrences with constant coefficients, signature (3).

The strict version is A336139.
Splittings of partitions are A323583.
Multiset partitions of partitions are A001970.
Partitions of each part of a partition are A063834.
Compositions of each part of a partition are A075900.
Strict partitions of each part of a strict partition are A279785.
Compositions of each part of a strict partition are A304961.
Strict compositions of each part of a composition are A307068.
Compositions of each part of a strict composition are A336127.
Cf. A000012, A000244, A000302, A001906, A006951, A011782, A022811, A071919.
Cf. A072706, A078008, A112467, A182105, A316245, A336128, A336135.
Cf. A339006, A355382, A355384, A355387.

[n eq 0 select 1 else 3^(n-1): n in [0..30]]; // G. C. Greubel, Nov 20 2023

a:= n-> ceil(3^(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2020

CoefficientList[Series[(1 - 2 x)/(1 - 3 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 3^(Range[0, 30])] (* G. C. Greubel, Nov 20 2023 *)

a(n)=max(1,3^(n-1)) \\ Charles R Greathouse IV, Jul 07 2011

Vec((1-2*x)/(1-3*x) + O(x^100)) \\  Altug Alkan, Oct 30 2015

[(3^n + 2*int(n==0))//3 for n in range(31)] # G. C. Greubel, Nov 20 2023

Binomial transform of A078008. - Paul Curtz, Aug 04 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: (1 - 2*x)/(1 - 3*x).
a(n) = A000244(n-1), n > 0. (End)
From Philippe Deléham, Nov 13 2008: (Start)
a(n) = Sum_{k=0..n} A112467(n,k)*2^k.
a(n) = Sum_{k=0..n} A071919(n,k)*2^k. (End)
Let A(x) be the g.f. Then B(x) = x*A(x) satisfies B(x/(1-x)) = x/(1 - 2*B(x)). - Vladimir Kruchinin, Dec 05 2011
G.f.: 1/(1 - (Sum_{k>=1} (x/(1 - x))^k)). - Joerg Arndt, Sep 30 2012
For n > 0, a(n) = 2*(Sum_{k=0..n-1} a(k)) - 1 = 3^(n-1). - J. Conrad, Oct 29 2015
G.f.: 1 + x/(1 + x)*(1 + 4*x/(1 + 4*x)*(1 + 7*x/(1 + 7*x)*(1 + 10*x/(1 + 10*x)*(1 + .... - Peter Bala, May 27 2017
Invert transform of A011782(n) = 2^(n-1). Second invert transform of A000012. - Gus Wiseman, Jul 19 2020
a(n) = ceiling(3^(n-1)). - Alois P. Heinz, Jul 26 2020
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: (2 + exp(3*x))/3.
a(n) = 3*a(n-1) for n > 1. (End)

Definition clarified by R. J. Mathar, Nov 11 2008

A072574 Triangle T(n,k) of number of compositions (ordered partitions) of n into exactly k distinct parts, 1<=k<=n.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 1, 4, 0, 0, 0, 1, 4, 6, 0, 0, 0, 1, 6, 6, 0, 0, 0, 0, 1, 6, 12, 0, 0, 0, 0, 0, 1, 8, 18, 0, 0, 0, 0, 0, 0, 1, 8, 24, 24, 0, 0, 0, 0, 0, 0, 1, 10, 30, 24, 0, 0, 0, 0, 0, 0, 0, 1, 10, 42, 48, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 48, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 60, 120, 0

Offset: 1

Henry Bottomley, Jun 21 2002

If terms in the compositions did not need to be distinct then the triangle would have values C(n-1,k-1), essentially A007318 offset.

			T(6,2)=4 since 6 can be written as 1+5=2+4=4+2=5+1.
Triangle starts (trailing zeros omitted for n>=10):
[ 1]  1;
[ 2]  1, 0;
[ 3]  1, 2, 0;
[ 4]  1, 2, 0, 0;
[ 5]  1, 4, 0, 0, 0;
[ 6]  1, 4, 6, 0, 0, 0;
[ 7]  1, 6, 6, 0, 0, 0, 0;
[ 8]  1, 6, 12, 0, 0, 0, 0, 0;
[ 9]  1, 8, 18, 0, 0, 0, 0, 0, 0;
[10]  1, 8, 24, 24, 0, 0, ...;
[11]  1, 10, 30, 24, 0, 0, ...;
[12]  1, 10, 42, 48, 0, 0, ...;
[13]  1, 12, 48, 72, 0, 0, ...;
[14]  1, 12, 60, 120, 0, 0, ...;
[15]  1, 14, 72, 144, 120, 0, 0, ...;
[16]  1, 14, 84, 216, 120, 0, 0, ...;
[17]  1, 16, 96, 264, 240, 0, 0, ...;
[18]  1, 16, 114, 360, 360, 0, 0, ...;
[19]  1, 18, 126, 432, 600, 0, 0, ...;
[20]  1, 18, 144, 552, 840, 0, 0, ...;
These rows (without the zeros) are shown in the Richmond/Knopfmacher reference.
From _Gus Wiseman_, Oct 17 2022: (Start)
Column n = 8 counts the following compositions.
  (8)  (1,7)  (1,2,5)
       (2,6)  (1,3,4)
       (3,5)  (1,4,3)
       (5,3)  (1,5,2)
       (6,2)  (2,1,5)
       (7,1)  (2,5,1)
              (3,1,4)
              (3,4,1)
              (4,1,3)
              (4,3,1)
              (5,1,2)
              (5,2,1)
(End)

Joerg Arndt, Table of n, a(n) for n = 1..5050 (rows 1..100, flattened).
B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97.
Index entries for sequences related to compositions

Columns (offset) include A057427 and A052928.
Row sums are A032020.
Cf. A060016, A072576.
A008289 is the version for partitions (zeros removed).
A072575 counts strict compositions by maximum.
A097805 is the non-strict version, or A007318 (zeros removed).
A113704 is the constant instead of strict version.
A216652 is a condensed version (zeros removed).
A336131 counts splittings of partitions with distinct sums.
A336139 counts strict compositions of each part of a strict composition.
Cf. A075900, A097910, A307068, A336127, A336128, A336130, A336132, A336141, A336142, A336342, A336343.

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],Length[#]==k&]],{n,0,15},{k,1,n}] (* Gus Wiseman, Oct 17 2022 *)

N=21;  q='q+O('q^N);
gf=sum(n=0,N, n! * z^n * q^((n^2+n)/2) / prod(k=1,n, 1-q^k ) );
/* print triangle: */
gf -= 1; /* remove row zero */
P=Pol(gf,'q);
{ for (n=1,N-1,
    p = Pol(polcoeff(P, n),'z);
    p += 'z^(n+1);  /* preserve trailing zeros */
    v = Vec(polrecip(p));
    v = vector(n,k,v[k]); /* trim to size n */
    print(v);
); }
/* Joerg Arndt, Oct 20 2012 */

T(n, k) = T(n-k, k)+k*T(n-k, k-1) [with T(n, 0)=1 if n=0 and 0 otherwise] = A000142(k)*A060016(n, k).

G.f.: sum(n>=0, n! * z^n * q^((n^2+n)/2) / prod(k=1..n, 1-q^k ) ), rows by powers of q, columns by powers of z; includes row 0 (drop term for n=0 for this triangle, see PARI code); setting z=1 gives g.f. for A032020. [Joerg Arndt, Oct 20 2012]

A336127 Number of ways to split a composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 2, 8, 16, 48, 144, 352, 896, 2432, 7168, 16896, 46080, 114688, 303104, 843776, 2080768, 5308416, 13762560, 34865152, 87818240, 241172480, 583008256, 1503657984, 3762290688, 9604956160, 23689428992, 60532195328, 156397207552, 385137770496, 967978254336

Offset: 0

Gus Wiseman, Jul 09 2020

nonn

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

			The a(0) = 1 through a(4) = 16 splits:
  ()  (1)  (2)    (3)        (4)
           (1,1)  (1,2)      (1,3)
                  (2,1)      (2,2)
                  (1,1,1)    (3,1)
                  (1),(2)    (1,1,2)
                  (2),(1)    (1,2,1)
                  (1),(1,1)  (1),(3)
                  (1,1),(1)  (2,1,1)
                             (3),(1)
                             (1,1,1,1)
                             (1),(1,2)
                             (1),(2,1)
                             (1,2),(1)
                             (2,1),(1)
                             (1),(1,1,1)
                             (1,1,1),(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A074854.
Starting with a strict composition gives A336128.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336130, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}]

a(n) = Sum_{k=0..n} 2^(n-k) k! A008289(n,k).

A336130 Number of ways to split a strict composition of n into contiguous subsequences all having the same sum.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 15, 13, 23, 27, 73, 65, 129, 133, 241, 375, 519, 617, 1047, 1177, 1859, 2871, 3913, 4757, 7653, 8761, 13273, 16155, 28803, 30461, 50727, 55741, 87743, 100707, 152233, 168425, 308937, 315973, 500257, 571743, 871335, 958265, 1511583, 1621273, 2449259, 3095511, 4335385, 4957877, 7554717, 8407537, 12325993, 14301411, 20348691, 22896077, 33647199, 40267141, 56412983, 66090291, 93371665, 106615841, 155161833

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

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

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with different instead of equal sums is A336128.
Starting with a non-strict composition gives A074854.
Starting with a partition gives A317715.
Starting with a strict partition gives A318683.
Set partitions with equal block-sums are A035470.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A271619, A279375, A305551, A317508, A318684, A326519, A336127, A336132, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],SameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]

a(31)-a(60) from Max Alekseyev, Feb 14 2024

A074854 a(n) = Sum_{d|n} (2^(n-d)).

Original entry on oeis.org

1, 3, 5, 13, 17, 57, 65, 209, 321, 801, 1025, 3905, 4097, 12417, 21505, 53505, 65537, 233985, 262145, 885761, 1327105, 3147777, 4194305, 16060417, 17825793, 50339841, 84148225, 220217345, 268435457, 990937089, 1073741825, 3506503681

Offset: 1

Miklos Kristof, Sep 11 2002

A034729 = Sum_{d|n} (2^(d-1)).
A055895 = 2*A034729.
If p is a prime, then a(p) = A034729(p) = 2^(p-1)+1.
From Gus Wiseman, Jul 14 2020: (Start)
Number of ways to tile a rectangle of size n using horizontal strips. Also the number of ways to choose a composition of each part of a constant partition of n. The a(0) = 1 through a(5) = 17 splittings are:
  ()  (1)  (2)      (3)          (4)              (5)
           (1,1)    (1,2)        (1,3)            (1,4)
           (1),(1)  (2,1)        (2,2)            (2,3)
                    (1,1,1)      (3,1)            (3,2)
                    (1),(1),(1)  (1,1,2)          (4,1)
                                 (1,2,1)          (1,1,3)
                                 (2,1,1)          (1,2,2)
                                 (2),(2)          (1,3,1)
                                 (1,1,1,1)        (2,1,2)
                                 (1,1),(2)        (2,2,1)
                                 (2),(1,1)        (3,1,1)
                                 (1,1),(1,1)      (1,1,1,2)
                                 (1),(1),(1),(1)  (1,1,2,1)
                                                  (1,2,1,1)
                                                  (2,1,1,1)
                                                  (1,1,1,1,1)
                                                  (1),(1),(1),(1),(1)
(End)

			Divisors of 6 = 1,2,3,6 and 6-1 = 5, 6-2 = 4, 6-3 = 3, 6-6 = 0. a(6) = 2^5 + 2^4 + 2^3 + 2^0 = 32 + 16 + 8 + 1 = 57.
G.f. = x + 3*x^2 + 5*x^3 + 13*x^4 + 17*x^5 + 57*x^6 + 65*x^7 + ...
a(14) = 1 + 2^7 + 2^12 + 2^13 = 12417. - _Gus Wiseman_, Jun 20 2018

Gus Wiseman, Table of n, a(n) for n = 1..32

Cf. A055895, A034729.
Cf. A080267.
Cf. A051731.
The version looking at lengths instead of sums is A101509.
The strictly increasing (or strictly decreasing) version is A304961.
Starting with a partition gives A317715.
Starting with a strict partition gives A318683.
Requiring distinct instead of equal sums gives A336127.
Starting with a strict composition gives A336130.
Partitions of partitions are A001970.
Splittings of compositions are A133494.
Splittings of partitions are A323583.
Cf. A006951, A063834, A075900, A279787, A305551, A316245, A323433, A336128.

a[ n_] := If[ n < 1, 0, Sum[ 2^(n - d), {d, Divisors[n]}]] (* Michael Somos, Mar 28 2013 *)

a(n)=if(n<1,0,2^n*polcoeff(sum(k=1,n,2/(2-x^k),x*O(x^n)),n))

a(n) = sumdiv(n,d, 2^(n-d) ); /* Joerg Arndt, Mar 28 2013 */

G.f.: 2^n times coefficient of x^n in Sum_{k>=1} x^k/(2-x^k). - Benoit Cloitre, Apr 21 2003; corrected by Joerg Arndt, Mar 28 2013
G.f.: Sum_{k>0} 2^(k-1)*x^k/(1-2^(k-1)*x^k). - Vladeta Jovovic, Jun 24 2003
G.f.: Sum_{n>=1} a*z^n/(1-a*z^n) (generalized Lambert series) where z=2*x and a=1/2. - Joerg Arndt, Jan 30 2011
Triangle A051731 mod 2 converted to decimal. - Philippe Deléham, Oct 04 2003
G.f.: Sum_{k>0} 1 / (2 / (2*x)^k - 1). - Michael Somos, Mar 28 2013

a(14) corrected from 9407 to 12417 by Gus Wiseman, Jun 20 2018

A336135 Number of ways to split an integer partition of n into contiguous subsequences with strictly decreasing sums.

Original entry on oeis.org

1, 1, 2, 5, 8, 16, 29, 50, 79, 135, 213, 337, 522, 796, 1191, 1791, 2603, 3799, 5506, 7873, 11154, 15768, 21986, 30565, 42218, 57917, 78968, 107399, 144932, 194889, 261061, 347773, 461249, 610059, 802778, 1053173, 1377325, 1793985, 2329009, 3015922, 3891142

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(5) = 16 splittings:
  (1)  (2)    (3)        (4)          (5)
       (1,1)  (2,1)      (2,2)        (3,2)
              (1,1,1)    (3,1)        (4,1)
              (2),(1)    (2,1,1)      (2,2,1)
              (1,1),(1)  (3),(1)      (3,1,1)
                         (1,1,1,1)    (3),(2)
                         (2,1),(1)    (4),(1)
                         (1,1,1),(1)  (2,1,1,1)
                                      (2,2),(1)
                                      (3),(1,1)
                                      (3,1),(1)
                                      (1,1,1,1,1)
                                      (2,1),(1,1)
                                      (2,1,1),(1)
                                      (1,1,1),(1,1)
                                      (1,1,1,1),(1)

Andrew Howroyd, Table of n, a(n) for n = 0..50

The version with equal sums is A317715.
The version with strictly increasing sums is A336134.
The version with weakly increasing sums is A336136.
The version with weakly decreasing sums is A316245.
The version with different sums is A336131.
Starting with a composition gives A304961.
Starting with a strict partition gives A318684.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A323433, A336128, A336130, A336133.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],Greater@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]

a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f, self()(r,min(m,t-1),t-1,0,0)) + self()(r,m-1,s,t,0) + if(t+m<=s, self()(r-m,min(m,r-m),s,t+m,1)))); recurse(n,n,n,0)} \\ Andrew Howroyd, Jan 18 2024

a(21) onwards from Andrew Howroyd, Jan 18 2024

A336342 Number of ways to choose a partition of each part of a strict composition of n.

Original entry on oeis.org

1, 1, 2, 7, 11, 29, 81, 155, 312, 708, 1950, 3384, 7729, 14929, 32407, 81708, 151429, 305899, 623713, 1234736, 2463743, 6208978, 10732222, 22487671, 43000345, 86573952, 160595426, 324990308, 744946690, 1336552491, 2629260284, 5050032692, 9681365777

Offset: 0

Gus Wiseman, Jul 18 2020

nonn

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

Is there a simple generating function?

			The a(1) = 1 through a(4) = 11 ways:
  (1)  (2)    (3)        (4)
       (1,1)  (2,1)      (2,2)
              (1,1,1)    (3,1)
              (1),(2)    (1),(3)
              (2),(1)    (2,1,1)
              (1),(1,1)  (3),(1)
              (1,1),(1)  (1,1,1,1)
                         (1),(2,1)
                         (2,1),(1)
                         (1),(1,1,1)
                         (1,1,1),(1)

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

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of partitions are A323583.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A011782, A304786, A316245, A317715, A336128, A336132, A336134, A336135.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

Table[Length[Join@@Table[Tuples[IntegerPartitions/@ctn],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,0,10}]

seq(n)={[subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, 1 + y*x^k*numbpart(k) + O(x*x^n)))]} \\ Andrew Howroyd, Apr 16 2021

G.f.: Sum_{k>=0} k! * [y^k](Product_{j>=1} 1 + y*x^j*A000041(j)). - Andrew Howroyd, Apr 16 2021

A336132 Number of ways to split a strict integer partition of n into contiguous subsequences all having different sums.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 11, 14, 21, 30, 37, 51, 66, 86, 120, 146, 186, 243, 303, 378, 495, 601, 752, 927, 1150, 1395, 1741, 2114, 2571, 3134, 3788, 4541, 5527, 6583, 7917, 9511, 11319, 13448, 16040, 18996, 22455, 26589, 31317, 36844, 43518, 50917, 59655, 69933

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(7) = 14 splits:
  (1)  (2)  (3)      (4)      (5)      (6)          (7)
            (2,1)    (3,1)    (3,2)    (4,2)        (4,3)
            (2),(1)  (3),(1)  (4,1)    (5,1)        (5,2)
                              (3),(2)  (3,2,1)      (6,1)
                              (4),(1)  (4),(2)      (4,2,1)
                                       (5),(1)      (4),(3)
                                       (3,2),(1)    (5),(2)
                                       (3),(2),(1)  (6),(1)
                                                    (4),(2,1)
                                                    (4,2),(1)
                                                    (4),(2),(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A318683.
Starting with a composition gives A336127.
Starting with a strict composition gives A336128.
Starting with a partition gives A336131.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336130, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]

A336131 Number of ways to split an integer partition of n into contiguous subsequences all having different sums.

Original entry on oeis.org

1, 1, 2, 6, 9, 20, 44, 74, 123, 231, 441, 681, 1188, 1889, 3110, 5448, 8310, 13046

Offset: 0

Gus Wiseman, Jul 11 2020

			The a(1) = 1 through a(4) = 9 splits:
  (1)  (2)    (3)        (4)
       (1,1)  (2,1)      (2,2)
              (1,1,1)    (3,1)
              (2),(1)    (2,1,1)
              (1),(1,1)  (3),(1)
              (1,1),(1)  (1,1,1,1)
                         (2,1),(1)
                         (1),(1,1,1)
                         (1,1,1),(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A317715.
Starting with a composition gives A336127.
Starting with a strict composition gives A336128.
Starting with a strict partition gives A336132.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A323433, A336130, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]

A336134 Number of ways to split an integer partition of n into contiguous subsequences with strictly increasing sums.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 17, 27, 37, 62, 82, 125, 168, 246, 320, 462, 585, 839, 1078, 1466, 1830, 2528, 3136, 4188, 5210, 6907, 8498, 11177, 13570, 17668, 21614, 27580, 33339, 42817, 51469, 65083, 78457, 98409, 117602, 147106, 174663, 217400, 259318, 319076, 377707

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(6) = 17 splits:
  (1)  (2)    (3)        (4)          (5)            (6)
       (1,1)  (2,1)      (2,2)        (3,2)          (3,3)
              (1,1,1)    (3,1)        (4,1)          (4,2)
              (1),(1,1)  (2,1,1)      (2,2,1)        (5,1)
                         (1,1,1,1)    (3,1,1)        (2,2,2)
                         (1),(1,1,1)  (2,1,1,1)      (3,2,1)
                                      (2),(2,1)      (4,1,1)
                                      (1,1,1,1,1)    (2,2,1,1)
                                      (2),(1,1,1)    (2),(2,2)
                                      (1),(1,1,1,1)  (3,1,1,1)
                                      (1,1),(1,1,1)  (2,1,1,1,1)
                                                     (2),(2,1,1)
                                                     (1,1,1,1,1,1)
                                                     (2),(1,1,1,1)
                                                     (1),(1,1,1,1,1)
                                                     (1,1),(1,1,1,1)
                                                     (1),(1,1),(1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..75

The version with equal sums is A317715.
The version with strictly decreasing sums is A336135.
The version with weakly decreasing sums is A316245.
The version with different sums is A336131.
Starting with a composition gives A304961.
Starting with a strict partition gives A336133.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A318684, A323433, A336128, A336130.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],Less@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]

a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f && r > t && t >= s, self()(r,m,t+1,0,0)) + self()(r,m-1,s,t,0) + self()(r-m,min(m,r-m), s,t+m,1))); recurse(n,n,0,0,0)} \\ Andrew Howroyd, Jan 18 2024

a(21) onwards from Andrew Howroyd, Jan 18 2024

cp's OEIS Frontend

A133494 Diagonal of the array of iterated differences of A047848.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

Extensions

A072574 Triangle T(n,k) of number of compositions (ordered partitions) of n into exactly k distinct parts, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A336127 Number of ways to split a composition of n into contiguous subsequences with different sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A336130 Number of ways to split a strict composition of n into contiguous subsequences all having the same sum.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A074854 a(n) = Sum_{d|n} (2^(n-d)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A336135 Number of ways to split an integer partition of n into contiguous subsequences with strictly decreasing sums.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A336342 Number of ways to choose a partition of each part of a strict composition of n.

Views

Author

Keywords