A374631 Number of integer compositions of n whose leaders of weakly increasing runs are identical.

1, 1, 2, 3, 6, 10, 19, 34, 63, 116, 218, 405, 763, 1436, 2714, 5127, 9718, 18422, 34968, 66397, 126168, 239820, 456027, 867325, 1649970, 3139288, 5973746, 11368487, 21636909, 41182648, 78389204, 149216039, 284046349, 540722066, 1029362133, 1959609449

Offset

0,3

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..750 (first 101 terms from John Tyler Rascoe)

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

Formula

G.f.: 1 + Sum_{i>0} A(x,i) where A(x,i) = 1/(1-x^i) * (x^i + Sum_{z>0} ( ((x^i)/(1-x^i) * (-1 + Product_{j>i} (1/(1-x^j))))^z )) is the g.f. for compositions of this kind with all leaders equal to i. - John Tyler Rascoe, Jul 25 2024

Example

The composition (1,3,1,4,1,2,2,1) has maximal weakly increasing subsequences ((1,3),(1,4),(1,2,2),(1)), with leaders (1,1,1,1), so is counted under a(15).
The a(0) = 1 through a(6) = 19 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (12)   (13)    (14)     (15)
                 (111)  (22)    (23)     (24)
                        (112)   (113)    (33)
                        (121)   (122)    (114)
                        (1111)  (131)    (123)
                                (1112)   (141)
                                (1121)   (222)
                                (1211)   (1113)
                                (11111)  (1122)
                                         (1131)
                                         (1212)
                                         (1221)
                                         (1311)
                                         (11112)
                                         (11121)
                                         (11211)
                                         (12111)
                                         (111111)

Mathematica

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

Prog

(PARI)
C_x(N) = {my(x='x+O('x^N), h=1+sum(i=1, N, 1/(1-x^i)*(x^i+sum(z=1, N-i+1, (x^i/(1-x^i)*(-1+(1/prod(j=i+1, N-i, 1-x^j))))^z)))); Vec(h)}
C_x(40) \\ John Tyler Rascoe, Jul 25 2024

Crossrefs

Ranked by A374633 = positions of identical rows in A374629 (sums A374630).

Types of runs (instead of weakly increasing):

- For leaders of identical runs we have A000005 for n > 0, ranks A272919.

- For leaders of anti-runs we have A374517, ranks A374519.

- For leaders of strictly increasing runs we have A374686, ranks A374685.

- For leaders of weakly decreasing runs we have A374742, ranks A374744.

- For leaders of strictly decreasing runs we have A374760, ranks A374759.

Types of run-leaders (instead of identical):

- For strictly decreasing leaders we appear to have A188920.

- For weakly decreasing leaders we appear to have A189076.

- For distinct leaders we have A374632, ranks A374768.

- For strictly increasing leaders we have A374634.

- For weakly increasing leaders we have A374635.

A003242 counts anti-run compositions.

A011782 counts compositions.

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

A274174 counts contiguous compositions, ranks A374249.

A335456 counts patterns matched by compositions.

A335548 counts non-contiguous compositions, ranks A374253.

A374637 counts compositions by sum of leaders of weakly increasing runs.

Cf. A000009, A106356, A124766, A238343, A261982, A333213, A373949, A374518, A374687, A374743, A374761.

Sequence in context: A291875 A227309 A368279 * A123916 A000693 A054178

Adjacent sequences: A374628 A374629 A374630 * A374632 A374633 A374634

Keyword

nonn

Author

Gus Wiseman, Jul 23 2024

Extensions

a(26) onwards from John Tyler Rascoe, Jul 25 2024

Status

approved