A371417 Triangle read by rows: T(n,k) is the number of complete compositions of n with k parts.

1, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 3, 1, 0, 0, 0, 3, 4, 1, 0, 0, 0, 6, 6, 5, 1, 0, 0, 0, 0, 16, 10, 6, 1, 0, 0, 0, 0, 12, 30, 15, 7, 1, 0, 0, 0, 0, 12, 35, 50, 21, 8, 1, 0, 0, 0, 0, 24, 50, 75, 77, 28, 9, 1, 0, 0, 0, 0, 0, 90, 126, 140, 112, 36, 10, 1

Offset: 0

Author

John Tyler Rascoe, Mar 23 2024

Keywords

• nonn
• easy
• tabl

Comments

A composition (ordered partition) is complete if the set of parts both covers an interval (is gap-free) and contains 1.

Examples

			The triangle begins:
    k=0  1  2  3   4   5   6   7   8   9  10
n=0:  1;
n=1:  0, 1;
n=2:  0, 0, 1;
n=3:  0, 0, 2, 1;
n=4:  0, 0, 0, 3,  1;
n=5:  0, 0, 0, 3,  4,  1;
n=6:  0, 0, 0, 6,  6,  5,  1;
n=7:  0, 0, 0, 0, 16, 10,  6,  1;
n=8:  0, 0, 0, 0, 12, 30, 15,  7,  1;
n=9:  0, 0, 0, 0, 12, 35, 50, 21,  8,  1;
n=10: 0, 0, 0, 0, 24, 50, 75, 77, 28,  9,  1;
...
For n = 5 there are a total of 8 complete compositions:
  T(5,3) = 3: (221), (212), (122)
  T(5,4) = 4: (2111), (1211), (1121), (1112)
  T(5,5) = 1: (11111)
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

A107428 counts gap-free compositions.

A251729 counts gap-free but not complete compositions.

Cf. A107429 (row sums give complete compositions of n), A000670 (column sums), A152947 (number of nonzero terms per column).

Cf. A066099, A124774, A373118.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0,
     `if`(i=0, t!, 0), `if`(i<1 or n (p-> seq(coeff(p, x, i), i=0..n))(add(b(n, i, 0), i=0..n)):
seq(T(n), n=0..12);  # Alois P. Heinz, Apr 03 2024

PARI

G(N)={ my(z='z+O('z^N)); Vec(sum(i=1,N,z^(i*(i+1)/2)*t^i*prod(j=1,i,sum(k=0,N, (z^(j*k)*t^k)/(k+1)!))))}
my(v=G(10)); for(n=0, #v, if(n<1,print([1]), my(p=v[n], r=vector(n+1)); for(k=0, n, r[k+1] =k!*polcoeff(p, k)); print(r)))

Formula

T(n,k) = k!*[z^n*t^k] Sum_{i>0} z^(i*(i+1)/2)*t^i * Product_{j=1..i} Sum_{k>=0} (z^(j*k)*t^k)/(k+1)!.