cp's OEIS Frontend

A376263 Number of strict integer compositions of n whose leaders of increasing runs are increasing.

%I A376263 #13 Sep 18 2024 20:23:10
%S A376263 1,1,1,2,2,3,5,6,8,11,18,21,30,38,52,77,96,126,167,217,278,402,488,
%T A376263 647,822,1073,1340,1747,2324,2890,3695,4690,5924,7469,9407,11718,
%U A376263 15405,18794,23777,29507,37188,45720,57404,70358,87596,110672,135329,167018,206761,254200,311920
%N A376263 Number of strict integer compositions of n whose leaders of increasing runs are increasing.
%C A376263 The leaders of increasing runs of a sequence are obtained by splitting it into maximal increasing subsequences and taking the first term of each.
%H A376263 Andrew Howroyd, <a href="/A376263/b376263.txt">Table of n, a(n) for n = 0..1000</a>
%H A376263 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A376263 a(n) = Sum_{k>=1} A008289(n,k)*A000110(k-1) for n > 0. - _Andrew Howroyd_, Sep 18 2024
%e A376263 The a(1) = 1 through a(9) = 11 compositions:
%e A376263  (1) (2) (3)   (4)   (5)   (6)     (7)     (8)     (9)
%e A376263          (1,2) (1,3) (1,4) (1,5)   (1,6)   (1,7)   (1,8)
%e A376263                      (2,3) (2,4)   (2,5)   (2,6)   (2,7)
%e A376263                            (1,2,3) (3,4)   (3,5)   (3,6)
%e A376263                            (1,3,2) (1,2,4) (1,2,5) (4,5)
%e A376263                                    (1,4,2) (1,3,4) (1,2,6)
%e A376263                                            (1,4,3) (1,3,5)
%e A376263                                            (1,5,2) (1,5,3)
%e A376263                                                    (1,6,2)
%e A376263                                                    (2,3,4)
%e A376263                                                    (2,4,3)
%t A376263 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@#&&Less@@First/@Split[#,Less]&]],{n,0,15}]
%o A376263 (PARI) \\ here Q(n) gives n-th row of A008289.
%o A376263 Q(n)={Vecrev(polcoef(prod(k=1, n, 1 + y*x^k, 1 + O(x*x^n)), n)/y)}
%o A376263 a(n)={if(n==0, 1, my(r=Q(n), s=Vec(serlaplace(exp(exp(x+O(x^#r))- 1)))); sum(k=1, #r, r[k]*s[k]))} \\ _Andrew Howroyd_, Sep 18 2024
%Y A376263 For less-greater or greater-less we have A294617.
%Y A376263 This is a strict case of A374688, weak version A374635.
%Y A376263 The strict less-greater version is A374689, weak version A189076.
%Y A376263 A003242 counts anti-run compositions, ranks A333489.
%Y A376263 A011782 counts compositions, strict A032020.
%Y A376263 A238130, A238279, A333755 count compositions by number of runs.
%Y A376263 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A376263 A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y A376263 Cf. A000110, A008289, A056823, A106356, A188920, A238343, A261982, A274174, A333213, A374634, A374683, A374698, A374763.
%K A376263 nonn
%O A376263 0,4
%A A376263 _Gus Wiseman_, Sep 18 2024
%E A376263 a(26) onwards from _Andrew Howroyd_, Sep 18 2024