A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

0, 0, 0, 0, 0, 1, 3, 9, 25, 63, 152, 355, 809, 1804, 3963, 8590, 18423, 39161, 82620, 173198, 361101, 749326, 1548609, 3189132, 6547190, 13404613, 27378579, 55801506, 113517749, 230544752, 467519136, 946815630, 1915199736, 3869892105, 7812086380, 15756526347

Offset: 0

Author

Gus Wiseman, Aug 06 2024

Keywords

• nonn

Comments

The leaders of maximal 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 composition y = (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), which are not weakly decreasing, so y is counted under a(12).
The a(0) = 0 through a(8) = 25 compositions:
  .  .  .  .  .  (122)  (132)   (133)    (143)
                        (1122)  (142)    (152)
                        (1221)  (1132)   (233)
                                (1222)   (1133)
                                (1321)   (1142)
                                (2122)   (1223)
                                (11122)  (1232)
                                (11221)  (1322)
                                (12211)  (1331)
                                         (1421)
                                         (2132)
                                         (3122)
                                         (11132)
                                         (11222)
                                         (11321)
                                         (12122)
                                         (12212)
                                         (12221)
                                         (13211)
                                         (21122)
                                         (21221)
                                         (111122)
                                         (111221)
                                         (112211)
                                         (122111)
		

Links

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

Crossrefs

For leaders of constant runs we have A056823.

For leaders of weakly increasing runs we have A374636, complement A189076?

The complement is counted by A374697.

For leaders of anti-runs we have A374699, complement A374682.

Other functional neighbors: A188920, A374764, A374765.

A003242 counts anti-run compositions, ranks A333489.

A011782 counts compositions.

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

A335456 counts patterns matched by compositions.

A373949 counts compositions by run-compressed sum, opposite A373951.

A374700 counts compositions by sum of leaders of strictly increasing runs.

Cf. A106356, A238343, A261982, A333213, A374632, A374679, A374683, A374689.

Programs

Mathematica

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

Formula

a(n) = A011782(n) - A374697(n). - Jinyuan Wang, Feb 13 2025

Extensions

More terms from Jinyuan Wang, Feb 13 2025