A374705 Number of integer compositions of n whose leaders of maximal strictly increasing runs sum to 2.

0, 0, 2, 0, 2, 3, 4, 7, 8, 14, 17, 27, 33, 48, 63, 84, 112, 147, 191, 248, 322, 409, 527, 666, 845, 1062, 1336, 1666, 2079, 2579, 3190, 3936, 4842, 5933, 7259, 8854, 10768, 13074, 15826, 19120, 23048, 27728, 33279, 39879, 47686, 56916, 67818, 80667, 95777, 113552, 134396

Offset

0,3

Comments

The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Formula

G.f.: (x*Q(x)/(1 + x))^2 + x^2*Q(x)/((1 + x)*(1 + x^2)), where Q(x) is the g.f. of A000009. - Andrew Howroyd, Aug 14 2024

Example

The a(0) = 0 through a(9) = 14 compositions:
  .  .  (2)   .  (112)  (23)   (24)    (25)    (26)    (27)
        (11)     (121)  (113)  (114)   (115)   (116)   (117)
                        (131)  (141)   (151)   (161)   (171)
                               (1212)  (1123)  (1124)  (234)
                                       (1213)  (1214)  (1125)
                                       (1231)  (1241)  (1134)
                                       (1312)  (1313)  (1215)
                                               (1412)  (1251)
                                                       (1314)
                                                       (1341)
                                                       (1413)
                                                       (1512)
                                                       (12123)
                                                       (12312)

Mathematica

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#, Less]]==2&]], {n, 0, 15}]

Prog

(PARI) seq(n)={my(A=O(x^(n-1)), q=eta(x^2 + A)/eta(x + A)); Vec((q*x/(1 + x))^2 + q*x^2/((1 + x)*(1 + x^2)), -n-1)} \\ Andrew Howroyd, Aug 14 2024

Crossrefs

For leaders of weakly decreasing runs we have A004526.

The case of strict compositions is A096749.

For leaders of anti-runs we have column k = 2 of A374521.

Leaders of strictly increasing runs in standard compositions are A374683.

Ranked by positions of 2s in A374684.

Column k = 2 of A374700.

A003242 counts anti-run compositions.

A011782 counts compositions.

A238130, A238279, A333755 count compositions by number of runs.

A274174 counts contiguous compositions, ranks A374249.

Cf. A000009, A096765, A106356, A124766, A238343, A261982, A333213.

Sequence in context: A279679 A261096 A257092 * A028376 A059235 A261216

Adjacent sequences: A374702 A374703 A374704 * A374706 A374707 A374708

Keyword

nonn

Author

Gus Wiseman, Aug 12 2024

Extensions

a(26) onwards from Andrew Howroyd, Aug 14 2024

Status

approved