cp's OEIS Frontend

A374690 Number of integer compositions of n whose leaders of strictly increasing runs are weakly increasing.

%I A374690 #10 Aug 08 2024 17:42:13
%S A374690 1,1,2,3,6,10,19,34,63,115,211,387,710,1302,2385,4372,8009,14671,
%T A374690 26867,49196,90069,164884,301812,552406,1011004,1850209,3385861,
%U A374690 6195832,11337470,20745337,37959030,69454669,127081111,232517129,425426211,778376479,1424137721
%N A374690 Number of integer compositions of n whose leaders of strictly increasing runs are weakly increasing.
%C A374690 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 A374690 Christian Sievers, <a href="/A374690/b374690.txt">Table of n, a(n) for n = 0..500</a>
%H A374690 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374690 The composition (1,1,3,2,3,2) has strictly increasing runs ((1),(1,3),(2,3),(2)), with leaders (1,1,2,2), so is counted under a(12).
%e A374690 The a(0) = 1 through a(6) = 19 compositions:
%e A374690   ()  (1)  (2)   (3)    (4)     (5)      (6)
%e A374690            (11)  (12)   (13)    (14)     (15)
%e A374690                  (111)  (22)    (23)     (24)
%e A374690                         (112)   (113)    (33)
%e A374690                         (121)   (122)    (114)
%e A374690                         (1111)  (131)    (123)
%e A374690                                 (1112)   (132)
%e A374690                                 (1121)   (141)
%e A374690                                 (1211)   (222)
%e A374690                                 (11111)  (1113)
%e A374690                                          (1122)
%e A374690                                          (1131)
%e A374690                                          (1212)
%e A374690                                          (1311)
%e A374690                                          (11112)
%e A374690                                          (11121)
%e A374690                                          (11211)
%e A374690                                          (12111)
%e A374690                                          (111111)
%t A374690 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],LessEqual@@First/@Split[#,Less]&]],{n,0,15}]
%Y A374690 Ranked by positions of weakly increasing rows in A374683.
%Y A374690 Types of runs (instead of strictly increasing):
%Y A374690 - For leaders of identical runs we have A000041.
%Y A374690 - For leaders of anti-runs we have A374681.
%Y A374690 - For leaders of weakly increasing runs we have A374635.
%Y A374690 - For leaders of weakly decreasing runs we have A188900.
%Y A374690 - For leaders of strictly decreasing runs we have A374764.
%Y A374690 Types of run-leaders (instead of weakly increasing):
%Y A374690 - For identical leaders we have A374686, ranks A374685.
%Y A374690 - For distinct leaders we have A374687, ranks A374698.
%Y A374690 - For strictly increasing leaders we have A374688.
%Y A374690 - For strictly decreasing leaders we have A374689.
%Y A374690 - For weakly decreasing leaders we have A374697.
%Y A374690 A003242 counts anti-run compositions, ranks A333489.
%Y A374690 A011782 counts compositions.
%Y A374690 A238130, A238279, A333755 count compositions by number of runs.
%Y A374690 A335456 counts patterns matched by compositions.
%Y A374690 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374690 A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y A374690 Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374629, A374630, A374632, A374679.
%K A374690 nonn
%O A374690 0,3
%A A374690 _Gus Wiseman_, Jul 27 2024
%E A374690 a(26) and beyond from _Christian Sievers_, Aug 08 2024