A374631 - OEIS

%I #21 Jul 26 2024 19:11:49

%S 1,1,2,3,6,10,19,34,63,116,218,405,763,1436,2714,5127,9718,18422,

%T 34968,66397,126168,239820,456027,867325,1649970,3139288,5973746,

%U 11368487,21636909,41182648,78389204,149216039,284046349,540722066,1029362133,1959609449

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

%C 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.

%H Alois P. Heinz, <a href="/A374631/b374631.txt">Table of n, a(n) for n = 0..750</a> (first 101 terms from John Tyler Rascoe)

%H Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%F 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

%e 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).

%e The a(0) = 1 through a(6) = 19 compositions:

%e   ()  (1)  (2)   (3)    (4)     (5)      (6)

%e            (11)  (12)   (13)    (14)     (15)

%e                  (111)  (22)    (23)     (24)

%e                         (112)   (113)    (33)

%e                         (121)   (122)    (114)

%e                         (1111)  (131)    (123)

%e                                 (1112)   (141)

%e                                 (1121)   (222)

%e                                 (1211)   (1113)

%e                                 (11111)  (1122)

%e                                          (1131)

%e                                          (1212)

%e                                          (1221)

%e                                          (1311)

%e                                          (11112)

%e                                          (11121)

%e                                          (11211)

%e                                          (12111)

%e                                          (111111)

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

%o (PARI)

%o 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)}

%o C_x(40) \\ _John Tyler Rascoe_, Jul 25 2024

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

%Y Types of runs (instead of weakly increasing):

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

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

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

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

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

%Y Types of run-leaders (instead of identical):

%Y - For strictly decreasing leaders we appear to have A188920.

%Y - For weakly decreasing leaders we appear to have A189076.

%Y - For distinct leaders we have A374632, ranks A374768.

%Y - For strictly increasing leaders we have A374634.

%Y - For weakly increasing leaders we have A374635.

%Y A003242 counts anti-run compositions.

%Y A011782 counts compositions.

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

%Y A274174 counts contiguous compositions, ranks A374249.

%Y A335456 counts patterns matched by compositions.

%Y A335548 counts non-contiguous compositions, ranks A374253.

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

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 23 2024

%E a(26) onwards from _John Tyler Rascoe_, Jul 25 2024