A072574 -id:A072574 - cp's OEIS frontend

A336141 Number of ways to choose a strict composition of each part of an integer partition of n.

1, 1, 2, 5, 9, 17, 41, 71, 138, 270, 518, 938, 1863, 3323, 6163, 11436, 20883, 37413, 69257, 122784, 221873, 397258, 708142, 1249955, 2236499, 3917628, 6909676, 12130972, 21251742, 36973609, 64788378, 112103360, 194628113, 336713377, 581527210, 1000153063

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) = 17 ways:
  (1)  (2)      (3)          (4)              (5)
       (1),(1)  (1,2)        (1,3)            (1,4)
                (2,1)        (3,1)            (2,3)
                (2),(1)      (2),(2)          (3,2)
                (1),(1),(1)  (3),(1)          (4,1)
                             (1,2),(1)        (3),(2)
                             (2,1),(1)        (4),(1)
                             (2),(1),(1)      (1,2),(2)
                             (1),(1),(1),(1)  (1,3),(1)
                                              (2,1),(2)
                                              (3,1),(1)
                                              (2),(2),(1)
                                              (3),(1),(1)
                                              (1,2),(1),(1)
                                              (2,1),(1),(1)
                                              (2),(1),(1),(1)
                                              (1),(1),(1),(1),(1)

Alois P. Heinz, Table of n, a(n) for n = 0..5623

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, 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

Table[Length[Join@@Table[Tuples[Join@@Permutations/@Select[IntegerPartitions[#],UnsameQ@@#&]&/@ctn],{ctn,IntegerPartitions[n]}]],{n,0,10}]
(* 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[n==0 || i==1, 1, g[n, i-1] +
     b[i, i, 0] g[n-i, Min[n-i, i]]];
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/(1 - A032020(k)*x^k).

A072576 Limit of number of compositions (ordered partitions) of m into distinct parts where largest part is exactly m-n, for m sufficiently large given n.

Original entry on oeis.org

1, 2, 2, 8, 8, 14, 38, 44, 68, 98, 242, 272, 440, 590, 878, 1772, 2180, 3194, 4466, 6320, 8432, 16190, 19262, 28580, 38276, 54314, 70730, 99152, 163328, 204230, 286670, 386132, 527132, 695978, 941738, 1220984, 1950128, 2390294, 3321398, 4342148, 5929532, 7616642, 10284410

Offset: 0

Henry Bottomley, Jun 21 2002

nonn

Consider an ordered 1 X n tiling of white tiles whose lengths are all distinct from each other, and whose sum is n. Now introduce into this tiling a red square. The resulting number of compositions is a(n). - Gregory L. Simay, Oct 25 2019

			a(3) = 8 because for any m > 6 the number of compositions of e.g. m=7 into distinct parts where the largest part is exactly m-3 = 7-3 = 4 is eight, since 7 can be written as 4+3 = 4+2+1 = 4+1+2 = 3+4 = 2+4+1 = 2+1+4 = 1+4+2 = 1+2+4.
Note that in the example immediately above, 4 corresponds to the red square, since it is greater than--and therefore distinct from--parts 1,2 and 3, which correspond to the distinct white tiles. More generally, for the compositions of n having all parts distinct, the red square must correspond to a positive integer > n in order for the number of resulting compositions to be a(n). - _Gregory L. Simay_, Oct 25 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Index entries for sequences related to compositions

Cf. A072575.

Cf. A032020. - Alois P. Heinz, Dec 12 2012

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0))) end:
a:= proc(n) local l; l:= b(n, n): add( i! * l[i], i=1..nops(l)) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 12 2012

b[n_, i_] := If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-1], If[i>n, {}, Prepend[b[n-i, i-1], 0]]}]]]; a[n_] := Module[{l}, l = b[n, n]; Sum[i!*l[[i]], {i, 1, Length[l]}]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 31 2014, after Alois P. Heinz *)

N=66;  q='q+O('q^N);
gf=sum(n=0,N, (n+1)!*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
Vec(gf)
/* Joerg Arndt, Oct 20 2012 */

a(n) = Sum_k (k+1)! * A060016(n,k) = Sum_k (k+1) * A072574(n,k).

a(n) = Sum_k (k+1)! * A008289(n,k). - Alois P. Heinz, Dec 12 2012

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).

A364908 Number of ways to write n as a nonnegative linear combination of an integer composition of n.

Original entry on oeis.org

1, 1, 4, 15, 70, 314, 1542, 7428, 36860, 182911, 917188, 4612480, 23323662, 118273428, 601762636, 3069070533, 15689123386, 80356953555, 412300910566, 2118715503962, 10902791722490, 56175374185014, 289766946825180, 1496239506613985, 7733302967423382

Offset: 0

Gus Wiseman, Aug 22 2023

nonn

A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

			The a(3) = 15 ways to write 3 as a nonnegative linear combination of an integer composition of 3:
  1*3  0*2+3*1  1*1+1*2  0*1+0*1+3*1
       1*2+1*1  3*1+0*2  0*1+1*1+2*1
                         0*1+2*1+1*1
                         0*1+3*1+0*1
                         1*1+0*1+2*1
                         1*1+1*1+1*1
                         1*1+2*1+0*1
                         2*1+0*1+1*1
                         2*1+1*1+0*1
                         3*1+0*1+0*1

Alois P. Heinz, Table of n, a(n) for n = 0..500

The case with no zero coefficients is A011782.
The version for partitions is A364907, strict A364910.
The strict case is A364909.
A000041 counts integer partitions, strict A000009.
A011782 counts compositions, strict A032020.
A097805 counts compositions by length, strict A072574.
A116861 = positive linear combinations of strict ptns of k, reverse A364916.
A365067 = nonnegative linear combinations of strict partitions of k.
A364912 = positive linear combinations of partitions of k.
A364916 = positive linear combinations of strict partitions of k.
Cf. A364350, A364839, A364906, A364911, A364914, A365002, A365004.

b:= proc(n, m) option remember; `if`(n=0, `if`(m=0, 1, 0),
      add(add(b(n-i, m-i*j), j=0..m/i), i=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 28 2024

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[n,ptn],{ptn,Join@@Permutations /@ IntegerPartitions[n]}]],{n,0,5}]

a(8)-a(24) from Alois P. Heinz, Jan 28 2024

A364909 Number of ways to write n as a nonnegative linear combination of a strict integer composition of n.

Original entry on oeis.org

1, 1, 1, 5, 5, 7, 51, 45, 89, 109, 709, 733, 1495, 1935, 3119, 13785, 16611, 29035, 44611, 68733, 95193, 372897, 435007, 781345, 1177181, 1866659, 2600537, 3906561, 12052631, 14610799, 25407653, 37652265, 59943351, 84060993, 128112805, 172172117, 480353257, 578740011

Offset: 0

Gus Wiseman, Aug 18 2023

nonn

A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

			The a(0) = 1 through a(5) = 7 ways:
  .  1*1  1*2  1*3      1*4      1*5
               0*2+3*1  0*3+4*1  0*4+5*1
               1*1+1*2  1*1+1*3  1*1+1*4
               1*2+1*1  1*3+1*1  1*2+1*3
               3*1+0*2  4*1+0*3  1*3+1*2
                                 1*4+1*1
                                 5*1+0*4

The case with no zero coefficients is A032020.
The version for partitions is A364907, strict A364910(n) = A364916(n,n).
The non-strict version is A364908.
A000041 counts integer partitions, strict A000009.
A011782 counts compositions, strict A032020.
A008284 counts partitions by length, strict A008289.
A097805 counts compositions by length, strict A072574.
Cf. A116861, A364350, A364839, A364906, A364912, A364914, A365002, A365004.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[n,ptn],{ptn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,0,5}]

from math import factorial
from sympy.utilities.iterables import partitions
def A364909(n):
    if n == 0: return 1
    aset = tuple(set(p) for p in partitions(n) if max(p.values(),default=0)==1)
    return sum(factorial(len(t)) for p in partitions(n) for t in aset if set(p).issubset(t)) # Chai Wah Wu, Sep 21 2023

a(18)-a(37) from Chai Wah Wu, Sep 21 2023

A282748 Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_i, x_j) = 1 for all i != j (where 1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 3, 4, 1, 1, 2, 9, 4, 5, 1, 1, 6, 3, 16, 5, 6, 1, 1, 4, 15, 4, 25, 6, 7, 1, 1, 6, 9, 28, 5, 36, 7, 8, 1, 1, 4, 21, 16, 45, 6, 49, 8, 9, 1, 1, 10, 9, 52, 25, 66, 7, 64, 9, 10, 1, 1, 4, 39, 16, 105, 36, 91, 8, 81, 10, 11, 1, 1, 12, 9, 100, 25, 186, 49, 120, 9, 100, 11, 12, 1, 1, 6, 45, 16, 205, 36, 301, 64, 153, 10, 121, 12, 13, 1

Offset: 1

N. J. A. Sloane, Mar 05 2017

See A101391 for the triangle T(n,k) = number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_1,x_2,...,x_k) = 1 (2 <= k <= n).

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  2,  3,   1;
  1,  4,  3,   4,   1;
  1,  2,  9,   4,   5,   1;
  1,  6,  3,  16,   5,   6,  1;
  1,  4, 15,   4,  25,   6,  7,   1;
  1,  6,  9,  28,   5,  36,  7,   8,  1;
  1,  4, 21,  16,  45,   6, 49,   8,  9,   1;
  1, 10,  9,  52,  25,  66,  7,  64,  9,  10,  1;
  1,  4, 39,  16, 105,  36, 91,   8, 81,  10, 11,  1;
  1, 12,  9, 100,  25, 186, 49, 120,  9, 100, 11, 12, 1;
  ...
From _Gus Wiseman_, Nov 12 2020: (Start)
Row n = 6 counts the following compositions:
  (6)  (15)  (114)  (1113)  (11112)  (111111)
       (51)  (123)  (1131)  (11121)
             (132)  (1311)  (11211)
             (141)  (3111)  (12111)
             (213)          (21111)
             (231)
             (312)
             (321)
             (411)
(End)

Alois P. Heinz, Rows n = 1..200, flattened (first 100 rows from Chai Wah Wu)
Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, Fibonacci Quart. 44 (2006), no. 4, 316-323.

A072704 counts the unimodal instead of coprime version.
A087087 and A335235 rank these compositions.
A101268 gives row sums.
A101391 is the relatively prime instead of pairwise coprime version.
A282749 is the unordered version.
A000740 counts relatively prime compositions, with strict case A332004.
A007360 counts pairwise coprime or singleton strict partitions.
A051424 counts pairwise coprime or singleton partitions, ranked by A302569.
A097805 counts compositions by sum and length.
A178472 counts compositions with a common divisor.
A216652 and A072574 count strict compositions by sum and length.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions, ranked by A302696.
A335235 ranks pairwise coprime or singleton compositions.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337562 counts pairwise coprime or singleton strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
Cf. A000837, A007359, A302568, A337461, A337561, A337667.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],Length[#]==1||CoprimeQ@@#&]],{n,10},{k,n}] (* Gus Wiseman, Nov 12 2020 *)

It seems that no general formula or recurrence is known, although Shonhiwa gives formulas for a few of the early diagonals.

A336140 Number of ways to choose a set partition of the parts of a strict integer composition of n.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 39, 43, 73, 107, 497, 531, 951, 1345, 2125, 8789, 9929, 16953, 24723, 38347, 52717, 219131, 240461, 419715, 600075, 938689, 1278409, 1928453, 6853853, 7815657, 13205247, 19051291, 29325121, 40353995, 60084905, 80722899, 277280079, 312239953

Offset: 0

Gus Wiseman, Jul 16 2020

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Set partitions are A000110.
Strict compositions are A032020.
Set partitions of binary indices are A050315.
Set partitions of strict partitions are A294617.
Cf. A000009, A001055, A008289, A035470, A063834, A072574, A137341, A279375.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 30 2020

Table[Sum[BellB[Length[ctn]],{ctn,Join@@Permutations/@Select[ IntegerPartitions[n],UnsameQ@@#&]}],{n,0,10}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0,
     BellB[p]*p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 40] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

a(n) = Sum_{k = 0..n} A000110(k) * A072574(n,k) = Sum_{k = 0..n} k! * A000110(k) * A008289(n,k).

A384907 Number of permutations of {1..n} with all distinct lengths of maximal anti-runs (not increasing by 1).

Original entry on oeis.org

1, 1, 1, 5, 17, 97, 587, 4291, 33109, 319967, 3106433, 35554459, 419889707, 5632467097, 77342295637, 1201240551077, 18804238105133, 328322081898745, 5832312989183807, 113154541564902427, 2229027473451951265, 47899977701182298255, 1037672943682453127645

Offset: 0

Gus Wiseman, Jun 21 2025

nonn

			The permutation (1,2,4,3,5,7,8,6,9) has maximal anti-runs ((1),(2,4,3,5,7),(8,6,9)), with lengths (1,5,3), so is counted under a(9).
The a(0) = 1 through a(4) = 17 permutations:
  ()  (1)  (2,1)  (1,3,2)  (1,2,4,3)
                  (2,1,3)  (1,3,2,4)
                  (2,3,1)  (1,4,2,3)
                  (3,1,2)  (1,4,3,2)
                  (3,2,1)  (2,1,3,4)
                           (2,1,4,3)
                           (2,3,1,4)
                           (2,4,1,3)
                           (2,4,3,1)
                           (3,1,4,2)
                           (3,2,1,4)
                           (3,2,4,1)
                           (3,4,2,1)
                           (4,1,3,2)
                           (4,2,1,3)
                           (4,3,1,2)
                           (4,3,2,1)

Christian Sievers, Table of n, a(n) for n = 0..449

For subsets instead of permutations we have A384177.
For strict partitions we have A384880, for runs A384178.
For partitions we have A384885, for runs A384884.
For runs instead of anti-runs we have A384891.
A010027 counts permutations by maximal anti-runs, for runs A123513.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000255, A044813, A072574, A242882, A325325, A328592, A329739, A351202, A356606, A384886, A384892.

Table[Length[Select[Permutations[Range[n]],UnsameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]

a(n)=if(n,my(b(n)=sum(i=0,n-1,(-1)^i*(n-i)!*binomial(n-1,i)), d=floor(sqrt(2*n)), p=polcoef(prod(i=1,n,1+x*y^i,1+O(y*y^n)*((1-x^(d+1))/(1-x))),n,y)); sum(i=1,d,b(n+1-i)*i!*polcoef(p,i)),1) \\ Christian Sievers, Jun 22 2025

a(n) = Sum_{k=1..n} ( T(n,k) * A000255(n-k) ) for n>=1, where T(n,k) is the number of compositions of n into k distinct parts (cf. A072574).

a(11) and beyond from Christian Sievers, Jun 22 2025

A362208 Irregular triangle read by rows: T(n, k) is the number of compositions (ordered partitions) of n into exactly k distinct parts between the members of [k^2].

Original entry on oeis.org

1, 0, 0, 2, 0, 2, 0, 4, 0, 2, 6, 0, 2, 6, 0, 0, 12, 0, 0, 18, 0, 0, 24, 24, 0, 0, 30, 24, 0, 0, 42, 48, 0, 0, 42, 72, 0, 0, 48, 120, 0, 0, 48, 144, 120, 0, 0, 48, 216, 120, 0, 0, 42, 264, 240, 0, 0, 42, 360, 360, 0, 0, 30, 432, 600, 0, 0, 24, 552, 840, 0, 0, 18, 648, 1200, 720

Offset: 1

Stefano Spezia, Apr 11 2023

			The irregular triangle begins:
    1;
    0;
    0, 2;
    0, 2;
    0, 4;
    0, 2,  6;
    0, 2,  6;
    0, 0, 12;
    0, 0, 18;
    0, 0, 24,  24;
    0, 0, 30,  24;
    0, 0, 42,  48;
    0, 0, 42,  72;
    0, 0, 48, 120;
    0, 0, 48, 144, 120;
    ...
T(7,3) = 6 since we have: 1+2+4, 1+4+2, 2+1+4, 2+4+1, 4+1+2, 4+2+1.

Cf. A000290, A003056 (row lengths), A072574, A216652.

Cf. A362209, A362221 (unordered partitions).

Flatten[Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n,All,Range[k^2]], UnsameQ@@#&], Length[#]==k&]], {n, 21}, {k, Floor[(Sqrt[8n+1]-1)/2]}]] (* After Gus Wiseman in A072574 *)

A362221 Irregular triangle read by rows: T(n, k) is the number of partitions of n into exactly k distinct parts between the members of [k^2].

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4, 1, 0, 0, 5, 1, 0, 0, 7, 2, 0, 0, 7, 3, 0, 0, 8, 5, 0, 0, 8, 6, 1, 0, 0, 8, 9, 1, 0, 0, 7, 11, 2, 0, 0, 7, 15, 3, 0, 0, 5, 18, 5, 0, 0, 4, 23, 7, 0, 0, 3, 27, 10, 1, 0, 0, 2, 34, 13, 1, 0, 0, 1, 38, 18, 2

Offset: 1

Stefano Spezia, Apr 11 2023

			The irregular triangle begins:
    1;
    0;
    0, 1;
    0, 1;
    0, 2;
    0, 1, 1;
    0, 1, 1;
    0, 0, 2;
    0, 0, 3;
    0, 0, 4, 1;
    0, 0, 5, 1;
    0, 0, 7, 2;
    0, 0, 7, 3;
    0, 0, 8, 5;
    0, 0, 8, 6, 1;
    ...
T(11,3) = 5 since we have: 1+2+8, 1+3+7, 1+4+6, 2+3+6, 2+4+5.

Cf. A000290, A003056 (row lengths), A072574, A216652, A362208 (compositions).

Flatten[Table[Length[Select[IntegerPartitions[n, All, Range[k^2]], UnsameQ@@# &&Length[#]==k&]], {n, 23}, {k, Floor[(Sqrt[8n+1]-1)/2]}]]

cp's OEIS Frontend

A336141 Number of ways to choose a strict composition of each part of an integer partition of n.

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A072576 Limit of number of compositions (ordered partitions) of m into distinct parts where largest part is exactly m-n, for m sufficiently large given 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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A364908 Number of ways to write n as a nonnegative linear combination of an integer composition of n.

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A364909 Number of ways to write n as a nonnegative linear combination of a strict integer composition of n.

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A282748 Triangle read by rows: T(n,k) is the number of compositions of n into k parts x_1, x_2, ..., x_k such that gcd(x_i, x_j) = 1 for all i != j (where 1 <= k <= n).

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A336140 Number of ways to choose a set partition of the parts of a strict integer composition of n.

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