cp's OEIS Frontend

A374697 Number of integer compositions of n whose leaders of strictly increasing runs are weakly decreasing.

%I A374697 #14 Jul 31 2024 22:11:36
%S A374697 1,1,2,4,8,15,29,55,103,193,360,669,1239,2292,4229,7794,14345,26375,
%T A374697 48452,88946,163187,299250,548543,1005172,1841418,3372603,6175853,
%U A374697 11307358,20699979,37890704,69351776,126926194,232283912,425075191,777848212,1423342837,2604427561
%N A374697 Number of integer compositions of n whose leaders of strictly increasing runs are weakly decreasing.
%C A374697 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.
%C A374697 Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the minima are weakly decreasing [weakly increasing works too].
%H A374697 Andrew Howroyd, <a href="/A374697/b374697.txt">Table of n, a(n) for n = 0..1000</a>
%H A374697 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374697 G.f.: 1/(Product_{k>=1} (1 - x^k*Product_{j>=k+1} (1 + x^j))). - _Andrew Howroyd_, Jul 31 2024
%e A374697 The composition (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), so is not counted under a(12).
%e A374697 The a(0) = 1 through a(5) = 15 compositions:
%e A374697   ()  (1)  (2)   (3)    (4)     (5)
%e A374697            (11)  (12)   (13)    (14)
%e A374697                  (21)   (22)    (23)
%e A374697                  (111)  (31)    (32)
%e A374697                         (112)   (41)
%e A374697                         (121)   (113)
%e A374697                         (211)   (131)
%e A374697                         (1111)  (212)
%e A374697                                 (221)
%e A374697                                 (311)
%e A374697                                 (1112)
%e A374697                                 (1121)
%e A374697                                 (1211)
%e A374697                                 (2111)
%e A374697                                 (11111)
%t A374697 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],GreaterEqual@@First/@Split[#,Less]&]],{n,0,15}]
%o A374697 (PARI) seq(n) = Vec(1/prod(k=1, n, 1 - x^k*prod(j=k+1, n-k, 1 + x^j, 1 + O(x^(n-k+1))))) \\ _Andrew Howroyd_, Jul 31 2024
%Y A374697 The opposite version is A374764.
%Y A374697 Ranked by positions of weakly decreasing rows in A374683.
%Y A374697 Interchanging weak/strict appears to give A188920, opposite A358836.
%Y A374697 Types of runs (instead of strictly increasing):
%Y A374697 - For leaders of identical runs we have A000041.
%Y A374697 - For leaders of anti-runs we have A374682.
%Y A374697 - For leaders of weakly increasing runs we have A189076, complement A374636.
%Y A374697 - For leaders of weakly decreasing runs we have A374747.
%Y A374697 - For leaders of strictly decreasing runs we have A374765.
%Y A374697 Types of run-leaders (instead of weakly decreasing):
%Y A374697 - For identical leaders we have A374686, ranks A374685.
%Y A374697 - For distinct leaders we have A374687, ranks A374698.
%Y A374697 - For weakly increasing leaders we have A374690.
%Y A374697 - For strictly increasing leaders we have A374688.
%Y A374697 - For strictly decreasing leaders we have A374689.
%Y A374697 A003242 counts anti-run compositions, ranks A333489.
%Y A374697 A011782 counts compositions.
%Y A374697 A238130, A238279, A333755 count compositions by number of runs.
%Y A374697 A335456 counts patterns matched by compositions.
%Y A374697 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374697 A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y A374697 Cf. A000009, A106356, A238343, A261982, A333213, A374632, A374679, A374740.
%K A374697 nonn
%O A374697 0,3
%A A374697 _Gus Wiseman_, Jul 27 2024
%E A374697 a(26) onwards from _Andrew Howroyd_, Jul 31 2024