A383713 Triangle read by rows: T(n,k) is the number of compositions of n with k parts all in standard order.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 3, 4, 1, 0, 0, 0, 0, 4, 6, 5, 1, 0, 0, 0, 0, 2, 10, 10, 6, 1, 0, 0, 0, 0, 1, 9, 20, 15, 7, 1, 0, 0, 0, 0, 1, 7, 25, 35, 21, 8, 1, 0, 0, 0, 0, 0, 7, 26, 55, 56, 28, 9, 1, 0, 0, 0, 0, 0, 4, 29, 71, 105, 84, 36, 10, 1

Offset: 0

Author

John Tyler Rascoe, May 06 2025

Keywords

• nonn
• easy
• tabl

Comments

A composition with parts in standard order satisfies the condition that for any part p > 1, the part p - 1 has already appeared. All compositions of this kind have first part 1.

Examples

			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, 1, 1],
 n=4  [0, 0, 0, 2, 1],
 n=5  [0, 0, 0, 1, 3,  1],
 n=6  [0, 0, 0, 1, 3,  4,  1],
 n=7  [0, 0, 0, 0, 4,  6,  5,  1],
 n=8  [0, 0, 0, 0, 2, 10, 10,  6,  1],
 n=9  [0, 0, 0, 0, 1,  9, 20, 15,  7, 1],
 n=10 [0, 0, 0, 0, 1,  7, 25, 35, 21, 8, 1],
 ...
Row n = 6 counts:
 T(6,3) = 1: (1,2,3).
 T(6,4) = 3: (1,1,2,2), (1,2,1,2), (1,2,2,1).
 T(6,5) = 4: (1,1,1,1,2), (1,1,1,2,1), (1,1,2,1,1), (1,2,1,1,1).
 T(6,6) = 1: (1,1,1,1,1,1).
		

Crossrefs

Cf. A000110 (column sums), A047998, A107429, A126347 (triangle transposed with no zeros), A278984, A383253 (row sums).

Programs

PARI

T_xy(max_row) = {my(N = max_row+1, x='x+O('x^N), h = 1 + sum(i=1,1+(N/2), y^i * x^(i*(i+1)/2)/prod(j=1,i, 1 - y*(x-x^(j+1))/(1-x)))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
T_xy(10)

Formula

G.f.: 1 + Sum_{i>0} y^i * x^(i*(i+1)/2) / Product_{j=1..i} 1 - y*(x - x^(j+1))/(1 - x).