cp's OEIS Frontend

A374761 Number of integer compositions of n whose leaders of strictly decreasing runs are distinct.

%I A374761 #11 Feb 14 2025 01:13:49
%S A374761 1,1,1,3,5,7,13,27,45,73,117,205,365,631,1061,1711,2777,4599,7657,
%T A374761 12855,21409,35059,56721,91149,146161,234981,379277,612825,988781,
%U A374761 1587635,2533029,4017951,6342853,9985087,15699577,24679859,38803005,60979839,95698257,149836255
%N A374761 Number of integer compositions of n whose leaders of strictly decreasing runs are distinct.
%C A374761 The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
%H A374761 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374761 The composition (3,1,4,3,2,1,2,8) has strictly decreasing runs ((3,1),(4,3,2,1),(2),(8)), with leaders (3,4,2,8), so is counted under a(24).
%e A374761 The a(0) = 1 through a(6) = 13 compositions:
%e A374761   ()  (1)  (2)  (3)   (4)    (5)    (6)
%e A374761                 (12)  (13)   (14)   (15)
%e A374761                 (21)  (31)   (23)   (24)
%e A374761                       (121)  (32)   (42)
%e A374761                       (211)  (41)   (51)
%e A374761                              (131)  (123)
%e A374761                              (311)  (132)
%e A374761                                     (141)
%e A374761                                     (213)
%e A374761                                     (231)
%e A374761                                     (312)
%e A374761                                     (321)
%e A374761                                     (411)
%t A374761 Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],UnsameQ@@First/@Split[#,Greater]&]],{n,0,15}]
%o A374761 (PARI) dfs(m, r, v) = 1 + sum(s=r, m, if(!setsearch(v, s), dfs(m-s, s, setunion(v, [s]))*x^s + sum(t=1, min(s-1, m-s), dfs(m-s-t, t, setunion(v, [s]))*x^(s+t)*prod(i=t+1, s-1, 1+x^i))));
%o A374761 lista(nn) = Vec(dfs(nn, 1, []) + O(x^(1+nn))); \\ _Jinyuan Wang_, Feb 13 2025
%Y A374761 For leaders of identical runs we have A274174, ranked by A374249.
%Y A374761 The weak opposite version is A374632, ranks A374768.
%Y A374761 The opposite version is A374687, ranks A374698.
%Y A374761 For identical instead of distinct leaders we have A374760, ranks A374759.
%Y A374761 The weak version is A374743, ranks A374701.
%Y A374761 Ranked by A374767.
%Y A374761 For partitions instead of compositions we have A375133.
%Y A374761 Other types of runs:
%Y A374761 - For leaders of identical runs we have A000005 for n > 0, ranks A272919.
%Y A374761 - For leaders of anti-runs we have A374518, ranked by A374638.
%Y A374761 Other types of run-leaders:
%Y A374761 - For strictly increasing leaders we have A374762.
%Y A374761 - For strictly decreasing leaders we have A374763.
%Y A374761 - For weakly increasing leaders we have A374764.
%Y A374761 - For weakly decreasing leaders we have A374765.
%Y A374761 A003242 counts anti-run compositions, ranks A333489.
%Y A374761 A011782 counts compositions.
%Y A374761 A238130, A238279, A333755 count compositions by number of runs.
%Y A374761 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374761 A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y A374761 Cf. A034296, A106356, A188920, A189076, A238343, A333213, A374517, A374631, A374640, A374686, A374742.
%K A374761 nonn
%O A374761 0,4
%A A374761 _Gus Wiseman_, Jul 29 2024
%E A374761 More terms from _Jinyuan Wang_, Feb 13 2025