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Number of integer compositions of n whose leaders of strictly increasing runs are weakly decreasing.
14

%I #14 Jul 31 2024 22:11:36

%S 1,1,2,4,8,15,29,55,103,193,360,669,1239,2292,4229,7794,14345,26375,

%T 48452,88946,163187,299250,548543,1005172,1841418,3372603,6175853,

%U 11307358,20699979,37890704,69351776,126926194,232283912,425075191,777848212,1423342837,2604427561

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

%C 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 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 Andrew Howroyd, <a href="/A374697/b374697.txt">Table of n, a(n) for n = 0..1000</a>

%H Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%F G.f.: 1/(Product_{k>=1} (1 - x^k*Product_{j>=k+1} (1 + x^j))). - _Andrew Howroyd_, Jul 31 2024

%e 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 The a(0) = 1 through a(5) = 15 compositions:

%e () (1) (2) (3) (4) (5)

%e (11) (12) (13) (14)

%e (21) (22) (23)

%e (111) (31) (32)

%e (112) (41)

%e (121) (113)

%e (211) (131)

%e (1111) (212)

%e (221)

%e (311)

%e (1112)

%e (1121)

%e (1211)

%e (2111)

%e (11111)

%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],GreaterEqual@@First/@Split[#,Less]&]],{n,0,15}]

%o (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 The opposite version is A374764.

%Y Ranked by positions of weakly decreasing rows in A374683.

%Y Interchanging weak/strict appears to give A188920, opposite A358836.

%Y Types of runs (instead of strictly increasing):

%Y - For leaders of identical runs we have A000041.

%Y - For leaders of anti-runs we have A374682.

%Y - For leaders of weakly increasing runs we have A189076, complement A374636.

%Y - For leaders of weakly decreasing runs we have A374747.

%Y - For leaders of strictly decreasing runs we have A374765.

%Y Types of run-leaders (instead of weakly decreasing):

%Y - For identical leaders we have A374686, ranks A374685.

%Y - For distinct leaders we have A374687, ranks A374698.

%Y - For weakly increasing leaders we have A374690.

%Y - For strictly increasing leaders we have A374688.

%Y - For strictly decreasing leaders we have A374689.

%Y A003242 counts anti-run compositions, ranks A333489.

%Y A011782 counts compositions.

%Y A238130, A238279, A333755 count compositions by number of runs.

%Y A335456 counts patterns matched by compositions.

%Y A373949 counts compositions by run-compressed sum, opposite A373951.

%Y A374700 counts compositions by sum of leaders of strictly increasing runs.

%Y Cf. A000009, A106356, A238343, A261982, A333213, A374632, A374679, A374740.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 27 2024

%E a(26) onwards from _Andrew Howroyd_, Jul 31 2024