cp's OEIS Frontend

A374687 Number of integer compositions of n whose leaders of strictly increasing runs are distinct.

%I A374687 #11 Feb 14 2025 01:14:25
%S A374687 1,1,1,3,3,7,11,15,27,45,65,101,161,251,381,573,865,1321,1975,2965,
%T A374687 4387,6467,9579,14091,20669,30135,43869,63531,91831,132575,190567,
%U A374687 273209,390659,557069,792371,1124381,1591977,2249029,3169993,4458163,6256201,8762251,12246541
%N A374687 Number of integer compositions of n whose leaders of strictly increasing runs are distinct.
%C A374687 The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
%H A374687 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374687 The a(0) = 1 through a(7) = 15 compositions:
%e A374687   ()  (1)  (2)  (3)   (4)   (5)    (6)    (7)
%e A374687                 (12)  (13)  (14)   (15)   (16)
%e A374687                 (21)  (31)  (23)   (24)   (25)
%e A374687                             (32)   (42)   (34)
%e A374687                             (41)   (51)   (43)
%e A374687                             (122)  (123)  (52)
%e A374687                             (212)  (132)  (61)
%e A374687                                    (213)  (124)
%e A374687                                    (231)  (133)
%e A374687                                    (312)  (142)
%e A374687                                    (321)  (214)
%e A374687                                           (241)
%e A374687                                           (313)
%e A374687                                           (412)
%e A374687                                           (421)
%t A374687 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],UnsameQ@@First/@Split[#,Less]&]],{n,0,15}]
%o A374687 (PARI) dfs(m, r, v) = 1 + sum(s=1, min(m, r), if(!setsearch(v, s), dfs(m-s, s, setunion(v, [s]))*x^s + sum(t=s+1, m-s, dfs(m-s-t, t, setunion(v, [s]))*x^(s+t)*prod(i=s+1, t-1, 1+x^i))));
%o A374687 lista(nn) = Vec(dfs(nn, nn, []) + O(x^(1+nn))); \\ _Jinyuan Wang_, Feb 13 2025
%Y A374687 Ranked by A374698.
%Y A374687 Types of runs (instead of strictly increasing):
%Y A374687 - For leaders of identical runs we have A274174 for n > 0, ranks A374249.
%Y A374687 - For leaders of anti-runs we have A374518, ranks A374638.
%Y A374687 - For leaders of weakly increasing runs we have A374632, ranks A374768.
%Y A374687 - For leaders of weakly decreasing runs we have A374743, ranks A374701.
%Y A374687 - For leaders of strictly decreasing runs we have A374761, ranks A374767.
%Y A374687 Types of run-leaders (instead of distinct):
%Y A374687 - For identical leaders we have A374686, ranks A374685.
%Y A374687 - For strictly increasing leaders we have A374688.
%Y A374687 - For strictly decreasing leaders we have A374689.
%Y A374687 - For weakly increasing leaders we have A374690.
%Y A374687 - For weakly decreasing leaders we have A374697.
%Y A374687 A003242 counts anti-run compositions, ranks A333489.
%Y A374687 A011782 counts compositions.
%Y A374687 A238130, A238279, A333755 count compositions by number of runs.
%Y A374687 A335456 counts patterns matched by compositions.
%Y A374687 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374687 A374683 lists leaders of strictly increasing runs of standard compositions.
%Y A374687 A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y A374687 Cf. A000009, A106356, A188920, A189076, A238343, A333213, A335548, A374629, A374634, A374635.
%K A374687 nonn
%O A374687 0,4
%A A374687 _Gus Wiseman_, Jul 27 2024
%E A374687 More terms from _Jinyuan Wang_, Feb 13 2025