# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a374632 Showing 1-1 of 1 %I A374632 #5 Jul 24 2024 09:22:23 %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 %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, Sequences counting and ranking compositions by their leaders (for six types of runs). %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}] %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,more %O A374632 0,3 %A A374632 _Gus Wiseman_, Jul 23 2024 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE