A374747 Number of integer compositions of n whose leaders of weakly decreasing runs are themselves weakly decreasing.

1, 1, 2, 3, 5, 8, 14, 24, 43, 76, 136, 242, 431, 764, 1353, 2387, 4202, 7376, 12918, 22567, 39338, 68421, 118765, 205743, 355756, 614038, 1058023, 1820029, 3125916, 5360659, 9179700, 15697559, 26807303, 45720739, 77881393, 132505599, 225182047, 382252310, 648187055

Offset: 0

Author

Gus Wiseman, Jul 26 2024

Keywords

• nonn

Comments

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

Examples

			The composition y = (3,2,1,2,2,1,2,5,1,1,1) has weakly decreasing runs ((3,2,1),(2,2,1),(2),(5,1,1,1)), with leaders (3,2,2,5), which are not weakly decreasing, so y is not counted under a(21).
The a(0) = 1 through a(6) = 14 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)
                                         (2112)
                                         (2121)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)
		

Links

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

Crossrefs

Ranked by positions of weakly decreasing rows in A374740, opposite A374629.

Types of runs (instead of weakly 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 strictly decreasing runs we have A374765.

Types of run-leaders (instead of weakly decreasing):

- For weakly increasing leaders we appear to have A188900.

- For identical leaders we have A374742, ranks A374744.

- For distinct leaders we have A374743, ranks A374701.

- For strictly increasing leaders we have opposite A374634.

- For strictly decreasing leaders we have A374746.

A011782 counts compositions.

A124765 counts weakly decreasing runs in standard 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.

A374748 counts compositions by sum of leaders of weakly decreasing runs.

Cf. A000009, A003242, A106356, A188920, A238343, A261982, A333213, A374630, A374635, A374636, A374741.

Programs

Mathematica

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

PARI

dfs(m, r, u) = 1 + sum(s=r+1, min(m, u), x^s/(1-x^s) + sum(t=1, min(s-1, m-s), dfs(m-s-t, t, s)*x^(s+t)/prod(i=t, s, 1-x^i)));
lista(nn) = Vec(dfs(nn, 0, nn) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 14 2025

Extensions

More terms from Jinyuan Wang, Feb 14 2025