cp's OEIS Frontend

A374632 Number of integer compositions of n whose leaders of weakly increasing runs are distinct.

%I A374632 #9 Feb 14 2025 01:13:54
%S A374632 1,1,2,4,7,13,23,40,69,119,200,335,557,917,1499,2433,3920,6280,10004,
%T A374632 15837,24946,39087,60952,94606,146203,224957,344748,526239,800251,
%U A374632 1212527,1830820,2754993,4132192,6178290,9209308,13686754,20282733,29973869,44175908,64936361
%N A374632 Number of integer compositions of n whose leaders of weakly increasing runs are distinct.
%C A374632 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 A374632 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374632 The composition (4,2,2,1,1,3) has weakly increasing runs ((4),(2,2),(1,1,3)), with leaders (4,2,1), so is counted under a(13).
%e A374632 The a(0) = 1 through a(5) = 13 compositions:
%e A374632   ()  (1)  (2)   (3)    (4)     (5)
%e A374632            (11)  (12)   (13)    (14)
%e A374632                  (21)   (22)    (23)
%e A374632                  (111)  (31)    (32)
%e A374632                         (112)   (41)
%e A374632                         (211)   (113)
%e A374632                         (1111)  (122)
%e A374632                                 (212)
%e A374632                                 (221)
%e A374632                                 (311)
%e A374632                                 (1112)
%e A374632                                 (2111)
%e A374632                                 (11111)
%t A374632 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],UnsameQ@@First/@Split[#,LessEqual]&]],{n,0,15}]
%o A374632 (PARI) dfs(m, r, v) = 1 + sum(s=1, min(m, r-1), if(!setsearch(v, s), dfs(m-s, s, setunion(v, [s]))*x^s/(1-x^s) + sum(t=s+1, m-s, dfs(m-s-t, t, setunion(v, [s]))*x^(s+t)/prod(i=s, t, 1-x^i))));
%o A374632 lista(nn) = Vec(dfs(nn, nn+1, []) + O(x^(1+nn))); \\ _Jinyuan Wang_, Feb 13 2025
%Y A374632 Ranked by A374768 = positions of distinct rows in A374629 (sums A374630).
%Y A374632 Types of runs (instead of weakly increasing):
%Y A374632 - For leaders of constant runs we have A274174, ranks A374249.
%Y A374632 - For leaders of anti-runs we have A374518, ranks A374638.
%Y A374632 - For leaders of strictly increasing runs we have A374687, ranks A374698.
%Y A374632 - For leaders of weakly decreasing runs we have A374743, ranks A335467.
%Y A374632 - For leaders of strictly decreasing runs we have A374761, ranks A374767.
%Y A374632 Types of run-leaders (instead of distinct):
%Y A374632 - For strictly decreasing leaders we appear to have A188920.
%Y A374632 - For weakly decreasing leaders we appear to have A189076.
%Y A374632 - For identical leaders we have A374631.
%Y A374632 - For weakly increasing leaders we have A374635.
%Y A374632 - For strictly increasing leaders we have A374634.
%Y A374632 A003242 counts anti-run compositions.
%Y A374632 A011782 counts compositions.
%Y A374632 A238130, A238279, A333755 count compositions by number of runs.
%Y A374632 A335456 counts patterns matched by compositions.
%Y A374632 A374637 counts compositions by sum of leaders of weakly increasing runs.
%Y A374632 Cf. A106356, A124766, A238343, A261982, A333213, A335548, A373949, A373953.
%K A374632 nonn
%O A374632 0,3
%A A374632 _Gus Wiseman_, Jul 23 2024
%E A374632 More terms from _Jinyuan Wang_, Feb 13 2025