A382312 Irregular triangle read by rows: T(n,k) is the number of compositions of n with k records.

1, 0, 1, 0, 2, 0, 3, 1, 0, 5, 3, 0, 8, 8, 0, 14, 17, 1, 0, 24, 36, 4, 0, 43, 72, 13, 0, 77, 143, 36, 0, 140, 281, 90, 1, 0, 256, 550, 213, 5, 0, 472, 1073, 484, 19, 0, 874, 2093, 1068, 61, 0, 1628, 4079, 2308, 177, 0, 3045, 7950, 4912, 476, 1, 0, 5719, 15498, 10328, 1217, 6

Offset: 0

Author

John Tyler Rascoe, Mar 21 2025

Keywords

• nonn
• tabf
• easy

Comments

A record in a composition is a part that is greater than all parts before it, reading left to right. The first part of any nonempty composition is considered a record.

Examples

			Triangle begins:
    k=0    1    2   3  4
 n= 0 1;
 n= 1 0,   1;
 n= 2 0,   2;
 n= 3 0,   3,   1;
 n= 4 0,   5,   3;
 n= 5 0,   8,   8;
 n= 6 0,  14,  17,  1;
 n= 7 0,  24,  36,  4;
 n= 8 0,  43,  72, 13;
 n= 9 0,  77, 143, 36;
 n=10 0, 140, 281, 90, 1;
 ...
The composition (2,1,1,2,4,2,1,5,7) has 4 records.
                 ^       ^     ^ ^
T(4,1) = 5 counts: (4), (3,1), (2,2), (2,1,1), (1,1,1,1).
T(4,2) = 3 counts: (1,1,2), (1,2,1), (1,1,3).
		

Links

• Alois P. Heinz, Rows n = 0..500, flattened (first 201 rows from John Tyler Rascoe)

Crossrefs

Cf. A002024 (row lengths), A011782 (row sums), A079500 (column k=1), A336482, A352525.

Programs

Maple

b:= proc(n, m) option remember; expand(`if`(n=0, 1, add(
      b(n-j, max(m, j))*`if`(j>m, x, 1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..16);  # Alois P. Heinz, Mar 28 2025

PARI

T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h=prod(i=1,N,1+y*x^i*(1-x)/(1-2*x+x^(i+1)))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
T_xy(12)

Formula

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

Sum_{k>0} T(n,k)*k = A336482(n).