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Number of integer compositions of n whose leaders of strictly decreasing runs are themselves strictly decreasing.
11

%I #14 Dec 30 2024 21:53:21

%S 1,1,1,2,3,4,6,10,15,22,32,47,71,106,156,227,328,473,683,986,1421,

%T 2040,2916,4149,5882,8314,11727,16515,23221,32593,45655,63810,88979,

%U 123789,171838,238055,329187,454451,626412,862164,1184917,1626124,2228324,3048982,4165640,5682847

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

%C The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

%H Andrew Howroyd, <a href="/A374763/b374763.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.: Sum_{k>=0} x^k*Q(k,x) where Q(0,x) = 1 and Q(k,x) = Q(k-1,x) * (1 + x^k*Product_{j=1..k} (1 + x^j)) for k > 0. - _Andrew Howroyd_, Dec 30 2024

%e The composition (3,1,2,1,1) has strictly decreasing runs ((3,1),(2,1),(1)), with leaders (3,2,1), so is counted under a(8).

%e The a(0) = 1 through a(8) = 15 compositions:

%e () (1) (2) (3) (4) (5) (6) (7) (8)

%e (21) (31) (32) (42) (43) (53)

%e (211) (41) (51) (52) (62)

%e (311) (312) (61) (71)

%e (321) (322) (413)

%e (411) (412) (422)

%e (421) (431)

%e (511) (512)

%e (3121) (521)

%e (3211) (611)

%e (3212)

%e (3221)

%e (4121)

%e (4211)

%e (31211)

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

%o (PARI) seq(n)={ my(A=O(x*x^n), p=1+A, q=p, r=p); for(k=1, n\2, r += x^k*q; p *= 1 + x^k; q *= 1 + x^k*p); Vec(r + x^(n\2+1)*q/(1-x)) } \\ _Andrew Howroyd_, Dec 30 2024

%Y The opposite version is A374688.

%Y The weak version is A374747.

%Y For partitions instead of compositions we have A375133.

%Y Other types of runs (instead of strictly decreasing):

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

%Y - For leaders of weakly increasing runs we appear to have A188920.

%Y - For leaders of anti-runs we have A374680.

%Y - For leaders of strictly increasing runs we have A374689.

%Y - For leaders of weakly decreasing runs we have A374746.

%Y Other types of run-leaders (instead of strictly decreasing):

%Y - For identical leaders we have A374760, ranks A374759.

%Y - For distinct leaders we have A374761, ranks A374767.

%Y - For strictly increasing leaders we have A374762.

%Y - For weakly increasing leaders we have A374764.

%Y - For weakly decreasing leaders we have A374765.

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

%Y A011782 counts compositions.

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

%Y A274174 counts contiguous compositions, ranks A374249.

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

%Y Cf. A000009, A106356, A188900, A238343, A261982, A333213, A374518, A374632, A374635, A374687, A374742, A374743.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jul 30 2024

%E a(24) onwards from _Andrew Howroyd_, Dec 30 2024