A374687 Number of integer compositions of n whose leaders of strictly increasing runs are distinct.

1, 1, 1, 3, 3, 7, 11, 15, 27, 45, 65, 101, 161, 251, 381, 573, 865, 1321, 1975, 2965, 4387, 6467, 9579, 14091, 20669, 30135, 43869, 63531, 91831, 132575, 190567, 273209, 390659, 557069, 792371, 1124381, 1591977, 2249029, 3169993, 4458163, 6256201, 8762251, 12246541

Offset: 0

Author

Gus Wiseman, Jul 27 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) = 1 through a(7) = 15 compositions:
  ()  (1)  (2)  (3)   (4)   (5)    (6)    (7)
                (12)  (13)  (14)   (15)   (16)
                (21)  (31)  (23)   (24)   (25)
                            (32)   (42)   (34)
                            (41)   (51)   (43)
                            (122)  (123)  (52)
                            (212)  (132)  (61)
                                   (213)  (124)
                                   (231)  (133)
                                   (312)  (142)
                                   (321)  (214)
                                          (241)
                                          (313)
                                          (412)
                                          (421)
		

Links

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

Crossrefs

Ranked by A374698.

Types of runs (instead of strictly increasing):

- For leaders of identical runs we have A274174 for n > 0, ranks A374249.

- For leaders of anti-runs we have A374518, ranks A374638.

- For leaders of weakly increasing runs we have A374632, ranks A374768.

- For leaders of weakly decreasing runs we have A374743, ranks A374701.

- For leaders of strictly decreasing runs we have A374761, ranks A374767.

Types of run-leaders (instead of distinct):

- For identical leaders we have A374686, ranks A374685.

- For strictly increasing leaders we have A374688.

- For strictly decreasing leaders we have A374689.

- For weakly increasing leaders we have A374690.

- For weakly decreasing leaders we have A374697.

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.

A374683 lists leaders of strictly increasing runs of standard compositions.

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

Cf. A000009, A106356, A188920, A189076, A238343, A333213, A335548, A374629, A374634, A374635.

Programs

Mathematica

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

PARI

dfs(m, r, v) = 1 + sum(s=1, min(m, r), if(!setsearch(v, s), dfs(m-s, s, setunion(v, [s]))*x^s + sum(t=s+1, m-s, dfs(m-s-t, t, setunion(v, [s]))*x^(s+t)*prod(i=s+1, t-1, 1+x^i))));
lista(nn) = Vec(dfs(nn, nn, []) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

Extensions

More terms from Jinyuan Wang, Feb 13 2025