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

Author

Gus Wiseman, Aug 12 2024

Keywords

• nonn

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.

Examples

			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)
		

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).

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.

Programs

Mathematica

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

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

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

Extensions

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