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A374705 Number of integer compositions of n whose leaders of maximal strictly increasing runs sum to 2.

%I A374705 #11 Aug 15 2024 02:04:32
%S A374705 0,0,2,0,2,3,4,7,8,14,17,27,33,48,63,84,112,147,191,248,322,409,527,
%T A374705 666,845,1062,1336,1666,2079,2579,3190,3936,4842,5933,7259,8854,10768,
%U A374705 13074,15826,19120,23048,27728,33279,39879,47686,56916,67818,80667,95777,113552,134396
%N A374705 Number of integer compositions of n whose leaders of maximal strictly increasing runs sum to 2.
%C A374705 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 A374705 Andrew Howroyd, <a href="/A374705/b374705.txt">Table of n, a(n) for n = 0..1000</a>
%H A374705 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A374705 G.f.: (x*Q(x)/(1 + x))^2 + x^2*Q(x)/((1 + x)*(1 + x^2)), where Q(x) is the g.f. of A000009. - _Andrew Howroyd_, Aug 14 2024
%e A374705 The a(0) = 0 through a(9) = 14 compositions:
%e A374705   .  .  (2)   .  (112)  (23)   (24)    (25)    (26)    (27)
%e A374705         (11)     (121)  (113)  (114)   (115)   (116)   (117)
%e A374705                         (131)  (141)   (151)   (161)   (171)
%e A374705                                (1212)  (1123)  (1124)  (234)
%e A374705                                        (1213)  (1214)  (1125)
%e A374705                                        (1231)  (1241)  (1134)
%e A374705                                        (1312)  (1313)  (1215)
%e A374705                                                (1412)  (1251)
%e A374705                                                        (1314)
%e A374705                                                        (1341)
%e A374705                                                        (1413)
%e A374705                                                        (1512)
%e A374705                                                        (12123)
%e A374705                                                        (12312)
%t A374705 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,Less]]==2&]],{n,0,15}]
%o A374705 (PARI) seq(n)={my(A=O(x^(n-1)), q=eta(x^2 + A)/eta(x + A)); Vec((q*x/(1 + x))^2 + q*x^2/((1 + x)*(1 + x^2)), -n-1)} \\ _Andrew Howroyd_, Aug 14 2024
%Y A374705 For leaders of weakly decreasing runs we have A004526.
%Y A374705 The case of strict compositions is A096749.
%Y A374705 For leaders of anti-runs we have column k = 2 of A374521.
%Y A374705 Leaders of strictly increasing runs in standard compositions are A374683.
%Y A374705 Ranked by positions of 2s in A374684.
%Y A374705 Column k = 2 of A374700.
%Y A374705 A003242 counts anti-run compositions.
%Y A374705 A011782 counts compositions.
%Y A374705 A238130, A238279, A333755 count compositions by number of runs.
%Y A374705 A274174 counts contiguous compositions, ranks A374249.
%Y A374705 Cf. A000009, A096765, A106356, A124766, A238343, A261982, A333213.
%K A374705 nonn
%O A374705 0,3
%A A374705 _Gus Wiseman_, Aug 12 2024
%E A374705 a(26) onwards from _Andrew Howroyd_, Aug 14 2024