A374765 Number of integer compositions of n whose leaders of strictly decreasing runs are weakly decreasing.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 88, 141, 225, 357, 565, 891, 1399, 2191, 3420, 5321, 8256, 12774, 19711, 30339, 46584, 71359, 109066, 166340, 253163, 384539, 582972, 882166, 1332538, 2009377, 3024969, 4546562, 6822926, 10223632, 15297051, 22855872, 34103117

Offset: 0

Author

Gus Wiseman, Jul 30 2024

Keywords

• nonn

Comments

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

Examples

			The composition (3,1,2,2,1) has strictly decreasing runs ((3,1),(2),(2,1)), with leaders (3,2,2), so is counted under a(9).
The a(0) = 1 through a(6) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (32)     (33)
                 (111)  (31)    (41)     (42)
                        (211)   (212)    (51)
                        (1111)  (221)    (222)
                                (311)    (312)
                                (2111)   (321)
                                (11111)  (411)
                                         (2121)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)
		

Links

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

Crossrefs

The opposite version is A374690.

Other types of runs (instead of strictly decreasing):

- For leaders of identical runs we have A000041.

- For leaders of weakly increasing runs we appear to have A189076.

- For leaders of anti-runs we have A374682.

- For leaders of strictly increasing runs we have A374697.

- For leaders of weakly decreasing runs we have A374747.

Other types of run-leaders (instead of weakly decreasing):

- For identical leaders we have A374760, ranks A374759.

- For distinct leaders we have A374761, ranks A374767.

- For strictly increasing leaders we have A374762.

- For strictly decreasing leaders we have A374763.

- For weakly increasing leaders we have A374764.

A003242 counts anti-run compositions, ranks A333489.

A011782 counts compositions.

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

A274174 counts contiguous compositions, ranks A374249.

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

Cf. A106356, A188900, A188920, A238343, A261982, A333213, A374635, A374636, A374689, A374742, A374743, A375133.

Programs

Mathematica

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

PARI

dfs(m, r, u) = 1 + sum(s=r, min(m, u), dfs(m-s, s, s)*x^s + sum(t=1, min(s-1, m-s), dfs(m-s-t, t, s)*x^(s+t)*prod(i=t+1, s-1, 1+x^i)));
lista(nn) = Vec(dfs(nn, 1, nn) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

Extensions

More terms from Jinyuan Wang, Feb 13 2025