A382288 Number of records in the n-th composition in standard order.

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1

Offset: 0

Author

John Tyler Rascoe, Mar 20 2025

Keywords

• nonn
• easy

Comments

Here a record is a part of the composition that is greater than all parts before it, reading left to right. The first part of any nonempty composition is a record so a(n) >= 1 for n > 0. See A066099 for the standard order of integer compositions.

The first appearance of k occurs at n = A164894(k) for k > 0.

Examples

			The 883rd composition is (1, 2, 1, 1, 3, 1, 1) with records 1, 2, and 3; so a(883) = 3.
                          ^  ^        ^
		

Crossrefs

Cf. A066099, A124762, A124767, A124768, A164894, A333381, A333382, A382312.

Programs

Python

def comp(n):
    return # see A357625
def A382288(n):
    r,c = 0,0
    for i in comp(n):
        if i > r:
            c += 1
            r = i
    return c

Formula

a(A164894(n)) = n for n > 0.