|
|
A374631
|
|
Number of integer compositions of n whose leaders of weakly increasing runs are identical.
|
|
25
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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)}
|
|
CROSSREFS
|
Types of runs (instead of weakly increasing):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- 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 strictly increasing leaders we have A374634.
- For weakly increasing leaders we have A374635.
A003242 counts anti-run compositions.
A335456 counts patterns matched by compositions.
A374637 counts compositions by sum of leaders of weakly increasing runs.
Cf. A000009, A106356, A124766, A238343, A261982, A333213, A373949, A374518, A374687, A374743, A374761.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|