A382312 -id:A382312 - cp's OEIS frontend

A336482 Total number of left-to-right maxima in all compositions of n.

0, 1, 2, 5, 11, 24, 51, 108, 226, 471, 976, 2015, 4146, 8508, 17418, 35590, 72597, 147868, 300797, 611202, 1240690, 2516268, 5099242, 10326282, 20897848, 42267257, 85442478, 172635651, 348651294, 703836046, 1420315254, 2865122304, 5777735296, 11647641296

Offset: 0

Alois P. Heinz, Jul 22 2020

nonn

			a(4) = 11: (1)111, (1)1(2), (1)(2)1, (2)11, (2)2, (1)(3), (3)1, (4).

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A000254 (the same for permutations of [n]), A225095, A336484, A336511, A336718, A382312.

b:= proc(n, m, c) option remember; `if`(n=0, c, add(
      b(n-j, max(m, j), c+`if`(j>m, 1, 0)), j=1..n))
    end:
a:= n-> b(n, -1, 0):
seq(a(n), n=0..50);

b[n_, m_, c_] := b[n, m, c] = If[n == 0, c, Sum[
     b[n - j, Max[m, j], c + If[j > m, 1, 0]], {j, 1, n}]];
a[n_] := b[n, -1, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)

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)))); h}
P_xy(N) = Pol(T_xy(N), {x})
B_x(N) = {my(cx = deriv(P_xy(N), y), y=1); Vecrev(eval(cx))}
B_x(30) \\ John Tyler Rascoe, Mar 22 2025

a(n) = Sum_{k>0} A382312(n,k)*k. - John Tyler Rascoe, Mar 22 2025

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

Original entry on oeis.org

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

John Tyler Rascoe, Mar 20 2025

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.

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

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

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

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

A383275 Number of compositions of n such that any part 1 can be k different colors where k is the current record having appeared in the composition.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 134, 454, 1634, 6245, 25321, 108779, 494443, 2374288, 12024257, 64100444, 358948674, 2106756217, 12931155910, 82823317389, 552400947902, 3829070637080, 27534807426150, 205066734143893, 1579309451332366, 12559941159979791, 103013928588389695

Offset: 0

John Tyler Rascoe, Apr 21 2025

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. A part 1 can be a record, iff it is the first part of a composition.

			a(3) = 5: (3), (1_a,2), (2,1_a), (2,1_b), (1_a,1_a,1_a).

Alois P. Heinz, Table of n, a(n) for n = 0..600

Cf. A000108, A011782, A088305, A382312, A382991, A383101, A383175.

b:= proc(n, m) option remember; `if`(n=0, 1, add(
      b(n-j, max(j, m))*`if`(j=1, m, 1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..26);  # Alois P. Heinz, Apr 23 2025

A_x(N) = {my(x='x+O('x^N)); Vec(prod(i=1,N,1+x^i/(1-i*x+(-x^2+x^(i+1))/(1-x))))}
A_x(30)

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

cp's OEIS Frontend

A336482 Total number of left-to-right maxima in all compositions of n.

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

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A383275 Number of compositions of n such that any part 1 can be k different colors where k is the current record having appeared in the composition.

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Programs

Maple

PARI

Formula