A336141 -id:A336141 - cp's OEIS frontend

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

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]

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

A307068 Expansion of 1/(1 - Sum_{k>=1} k!x^(k(k+1)/2) / Product_{j=1..k} (1 - x^j)).

Original entry on oeis.org

1, 1, 2, 6, 14, 34, 88, 216, 532, 1322, 3290, 8142, 20192, 50080, 124144, 307878, 763474, 1893038, 4694060, 11639580, 28861736, 71567206, 177460750, 440037738, 1091134276, 2705618900, 6708953156, 16635775698, 41250705518, 102286806130, 253634237896, 628921097352, 1559496588628

Offset: 0

Ilya Gutkovskiy, Mar 22 2019

nonn

Invert transform of A032020.
Number of ways to choose a strict composition of each part of a composition of n. - Gus Wiseman, Jul 18 2020
The Invert transform T(a) of a sequence a is given by T(a)n = Sum_c Product_i a(c_i), where the sum is over all compositions c of n. - _Gus Wiseman, Aug 01 2020

			From _Gus Wiseman_, Jul 18 2020: (Start)
The a(1) = 1 through a(4) = 14 ways to choose a strict composition of each part of a composition:
    (1)  (2)      (3)          (4)
         (1),(1)  (1,2)        (1,3)
                  (2,1)        (3,1)
                  (1),(2)      (1),(3)
                  (2),(1)      (2),(2)
                  (1),(1),(1)  (3),(1)
                               (1),(1,2)
                               (1),(2,1)
                               (1,2),(1)
                               (2,1),(1)
                               (1),(1),(2)
                               (1),(2),(1)
                               (2),(1),(1)
                               (1),(1),(1),(1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2536

The version for partitions is A270995.
Starting with a strict composition gives A336139.
Strict compositions are counted by A032020.
Partitions of each part of a partition are A063834.
Compositions of each part of a partition are A075900.
Compositions of each part of a composition are A133494.
Strict partitions of each part of a strict partition are A279785.
Compositions of each part of a strict partition are A304961.
Strict partitions of each part of a composition are A304969.
Compositions of each part of a strict composition are A336127.
Set partitions of strict compositions are A336140.
Strict compositions of each part of a partition are A336141.
Cf. A001970, A307067, A317536, A318683, A318684, A319794, A323583, A336128, A336130, A336132.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(1 - (&+[Factorial(k)*x^Binomial(k+1,2)/(&*[ 1-x^j: j in [1..k]]): k in [1..m+2]]) ) )); // G. C. Greubel, Jan 25 2024

T:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), T(n-k, k) +k*T(n-k, k-1)))
    end:
g:= proc(n) option remember; add(T(n, k), k=0..floor((sqrt(8*n+1)-1)/2)) end:
a:= proc(n) option remember; `if`(n<1, 1,
      add(a(n-i)*g(i), i=1..n))
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, Dec 16 2022

nmax = 32; CoefficientList[Series[1/(1 - Sum[k!*x^(k*(k+1)/2)/Product[ (1-x^j), {j,k}], {k,nmax}]), {x, 0, nmax}], x]

m=80;
def p(x, j): return product(1-x^k for k in range(1,j+1))
def f(x): return 1/(1 - sum(factorial(j)*x^binomial(j+1,2)/p(x,j) for j in range(1, m+3)) )
def A307068_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307068_list(m) # G. C. Greubel, Jan 25 2024

a(0) = 1; a(n) = Sum_{k=1..n} A032020(k)*a(n-k).

A336343 Number of ways to choose a strict partition of each part of a strict composition of n.

Original entry on oeis.org

1, 1, 1, 4, 6, 11, 26, 39, 78, 142, 320, 488, 913, 1558, 2798, 5865, 9482, 16742, 28474, 50814, 82800, 172540, 266093, 472432, 790824, 1361460, 2251665, 3844412, 7205416, 11370048, 19483502, 32416924, 54367066, 88708832, 149179800, 239738369, 445689392

Offset: 0

Gus Wiseman, Jul 19 2020

nonn

A strict composition of n (A032020) is a finite sequence of distinct positive integers summing to n.

Is there a simple generating function?

			The a(1) = 1 through a(5) = 11 ways:
  (1)  (2)  (3)      (4)        (5)
            (2,1)    (3,1)      (3,2)
            (1),(2)  (1),(3)    (4,1)
            (2),(1)  (3),(1)    (1),(4)
                     (1),(2,1)  (2),(3)
                     (2,1),(1)  (3),(2)
                                (4),(1)
                                (1),(3,1)
                                (2,1),(2)
                                (2),(2,1)
                                (3,1),(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 strict partitions are A072706.
Set partitions of strict partitions are A294617.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A011782, A304786, A318683, A318684, A319794, A323583, A336128, A336130, A336132, A336133.
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.

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

\\ here Q(N) gives A000009 as a vector.
Q(n) = {Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)))}
seq(n)={my(b=Q(n)); [subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, 1 + y*x^k*b[1+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*A000009(j)). - Andrew Howroyd, Apr 16 2021

A336142 Number of ways to choose a strict composition of each part of a strict integer partition of n.

Original entry on oeis.org

1, 1, 1, 4, 6, 11, 22, 41, 72, 142, 260, 454, 769, 1416, 2472, 4465, 7708, 13314, 23630, 40406, 68196, 119646, 203237, 343242, 586508, 993764, 1677187, 2824072, 4753066, 7934268, 13355658, 22229194, 36945828, 61555136, 102019156, 168474033, 279181966

Offset: 0

Gus Wiseman, Jul 18 2020

nonn

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

			The a(1) = 1 through a(5) = 11 ways:
  (1)  (2)  (3)      (4)        (5)
            (1,2)    (1,3)      (1,4)
            (2,1)    (3,1)      (2,3)
            (2),(1)  (3),(1)    (3,2)
                     (1,2),(1)  (4,1)
                     (2,1),(1)  (3),(2)
                                (4),(1)
                                (1,2),(2)
                                (1,3),(1)
                                (2,1),(2)
                                (3,1),(1)

Alois P. Heinz, Table of n, a(n) for n = 0..7725

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, A316245, A318684, A319794, A336128, A336130, A336132, 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.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 g(n$2):
seq(a(n), n=0..38);  # Alois P. Heinz, Jul 31 2020

strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[Join@@Permutations/@strptn[#]&/@ctn],{ctn,strptn[n]}]],{n,0,20}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
     If[n == 0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0,
     If[n == 0, 1, g[n, i-1] + b[i, i, 0]*g[n-i, Min[n-i, i-1]]]];
a[n_] := g[n, n];
a /@ Range[0, 38] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

G.f.: Product_{k >= 1} (1 + A032020(k)*x^k).

cp's OEIS Frontend

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

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

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A307068 Expansion of 1/(1 - Sum_{k>=1} k!*x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^j)).

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A336343 Number of ways to choose a strict partition of each part of a strict composition of n.

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A336142 Number of ways to choose a strict composition of each part of a strict integer partition of n.

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A307068 Expansion of 1/(1 - Sum_{k>=1} k!x^(k(k+1)/2) / Product_{j=1..k} (1 - x^j)).