cp's OEIS Frontend

A374742 Number of integer compositions of n whose leaders of weakly decreasing runs are identical.

%I A374742 #21 Apr 30 2025 09:16:47
%S A374742 1,1,2,3,5,8,13,21,34,54,87,138,220,349,556,881,1403,2229,3551,5653,
%T A374742 9019,14387,22988,36739,58785,94100,150765,241658,387617,622002,
%U A374742 998658,1604032,2577512,4143243,6662520,10716931,17243904,27753518,44680121,71947123,115880662
%N A374742 Number of integer compositions of n whose leaders of weakly decreasing runs are identical.
%C A374742 The weakly decreasing run-leaders of a sequence are obtained by splitting into maximal weakly decreasing subsequences and taking the first term of each.
%H A374742 John Tyler Rascoe, <a href="/A374742/b374742.txt">Table of n, a(n) for n = 0..200</a>
%H A374742 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374742 G.f.: 1 + Sum_{i>0} -1 + (1 + x^i/(1 - x^i))/(1 - B(i,x)) where B(i,x) = x^i/(1 - x^i) * Sum_{j=1..i-1} x^j * Product_{k=1..j} (1 - x^k)^(-1). - _John Tyler Rascoe_, Apr 29 2025
%e A374742 The composition (3,1,3,2,1,3,3) has maximal weakly decreasing subsequences ((3,1),(3,2,1),(3,3)), with leaders (3,3,3), so is counted under a(16).
%e A374742 The a(0) = 1 through a(6) = 13 compositions:
%e A374742   ()  (1)  (2)   (3)    (4)     (5)      (6)
%e A374742            (11)  (21)   (22)    (32)     (33)
%e A374742                  (111)  (31)    (41)     (42)
%e A374742                         (211)   (212)    (51)
%e A374742                         (1111)  (221)    (222)
%e A374742                                 (311)    (321)
%e A374742                                 (2111)   (411)
%e A374742                                 (11111)  (2112)
%e A374742                                          (2121)
%e A374742                                          (2211)
%e A374742                                          (3111)
%e A374742                                          (21111)
%e A374742                                          (111111)
%t A374742 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],SameQ@@First/@Split[#,GreaterEqual]&]],{n,0,15}]
%o A374742 (PARI)
%o A374742 B(i) = x^i/(1-x^i) * sum(j=1,i-1, x^j*prod(k=1,j, (1-x^k)^(-1)))
%o A374742 A_x(N) = {my(x='x+O('x^N)); Vec(1+sum(i=1,N,-1+(1+x^i/(1-x^i))/(1-B(i))))}
%o A374742 A_x(30) \\ _John Tyler Rascoe_, Apr 29 2025
%Y A374742 Ranked by A374744 = positions of identical rows in A374740, cf. A374629.
%Y A374742 Types of runs (instead of weakly decreasing):
%Y A374742 - For leaders of identical runs we have A000005 for n > 0, ranks A272919.
%Y A374742 - For leaders of anti-runs we have A374517, ranks A374519.
%Y A374742 - For leaders of strictly increasing runs we have A374686, ranks A374685.
%Y A374742 - For leaders of weakly increasing runs we have A374631, ranks A374633.
%Y A374742 - For leaders of strictly decreasing runs we have A374760, ranks A374759.
%Y A374742 Types of run-leaders (instead of identical):
%Y A374742 - For strictly decreasing leaders we have A374746.
%Y A374742 - For weakly decreasing leaders we have A374747.
%Y A374742 - For distinct leaders we have A374743, ranks A374701.
%Y A374742 - For weakly increasing leaders we appear to have A188900.
%Y A374742 A003242 counts anti-run compositions, ranks A333489.
%Y A374742 A011782 counts compositions.
%Y A374742 A238130, A238279, A333755 count compositions by number of runs.
%Y A374742 A274174 counts contiguous compositions, ranks A374249.
%Y A374742 A335456 counts patterns matched by compositions.
%Y A374742 A335548 counts non-contiguous compositions, ranks A374253.
%Y A374742 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374742 A374748 counts compositions by sum of leaders of weakly decreasing runs.
%Y A374742 Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632, A374634, A374635, A374741.
%K A374742 nonn
%O A374742 0,3
%A A374742 _Gus Wiseman_, Jul 25 2024
%E A374742 a(24)-a(40) from _Alois P. Heinz_, Jul 26 2024