This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A188920 #41 Apr 25 2025 23:40:01
%S A188920 1,1,2,4,7,13,22,38,63,105,169,274,434,686,1069,1660,2548,3897,5906,
%T A188920 8911,13352,19917,29532,43605,64056,93715,136499,198059,286233,412199,
%U A188920 591455,845851,1205687,1713286,2427177,3428611,4829563,6784550,9505840,13284849
%N A188920 a(n) is the limiting term of the n-th column of the triangle in A188919.
%C A188920 Also the number of integer compositions of n whose reverse avoids 12-1 and 23-1.
%C A188920 Theorem: The reverse of a composition avoids 12-1 and 23-1 iff its leaders of maximal weakly increasing runs, obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each, are strictly decreasing. For example, the composition y = (4,5,3,2,2,3,1,3,5) has reverse (5,3,1,3,2,2,3,5,4), which avoids 12-1 and 23-1, while the maximal weakly increasing runs of y are ((4,5),(3),(2,2,3),(1,3,5)), with leaders (4,3,2,1), which are strictly decreasing, as required. - _Gus Wiseman_, Aug 20 2024
%H A188920 John Tyler Rascoe, <a href="/A188920/b188920.txt">Table of n, a(n) for n = 0..200</a>
%H A188920 A. M. Baxter, <a href="https://pdfs.semanticscholar.org/2c5d/79e361d3aecb25c380402144177ad7cd9dc8.pdf">Algorithms for Permutation Statistics</a>, Ph. D. Dissertation, Rutgers University, May 2011.
%H A188920 Andrew M. Baxter and Lara K. Pudwell, <a href="http://arxiv.org/abs/1108.2642">Enumeration schemes for dashed patterns</a>, arXiv preprint arXiv:1108.2642 [math.CO], 2011-2012.
%H A188920 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>.
%H A188920 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A188920 a(n) = 2^(n-1) - A375140(n).
%F A188920 G.f.: 1 + Sum_{i>0} (B(i,x) * Product_{j=1..i-1} (1 + B(j,x))) where B(i,x) = (x^i)/(1-x^i) * Product_{j>i} (1/(1-x^j)). - _John Tyler Rascoe_, Aug 23 2024
%e A188920 From _Gus Wiseman_, Aug 20 2024: (Start)
%e A188920 The a(0) = 1 through a(6) = 22 compositions:
%e A188920 () (1) (2) (3) (4) (5) (6)
%e A188920 (11) (12) (13) (14) (15)
%e A188920 (21) (22) (23) (24)
%e A188920 (111) (31) (32) (33)
%e A188920 (112) (41) (42)
%e A188920 (211) (113) (51)
%e A188920 (1111) (122) (114)
%e A188920 (212) (123)
%e A188920 (221) (132)
%e A188920 (311) (213)
%e A188920 (1112) (222)
%e A188920 (2111) (312)
%e A188920 (11111) (321)
%e A188920 (411)
%e A188920 (1113)
%e A188920 (1122)
%e A188920 (2112)
%e A188920 (2211)
%e A188920 (3111)
%e A188920 (11112)
%e A188920 (21111)
%e A188920 (111111)
%e A188920 (End)
%t A188920 b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1, Sum[b[u - j, o + j - 1]*x^(o + j - 1), {j, 1, u}] + Sum[If[u == 0, b[u + j - 1, o - j]*x^(o - j), 0], {j, 1, o}]]];
%t A188920 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[0, n]];
%t A188920 Take[T[40], 40] (* _Jean-François Alcover_, Sep 15 2018, after _Alois P. Heinz_ in A188919 *)
%t A188920 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Greater@@First/@Split[Reverse[#],LessEqual]&]],{n,0,15}] (* _Gus Wiseman_, Aug 20 2024 *)
%t A188920 - or -
%t A188920 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MatchQ[#,{___,y_,z_,___,x_,___}/;x<=y<z]&]],{n,0,15}] (* _Gus Wiseman_, Aug 20 2024 *)
%o A188920 (PARI)
%o A188920 B_x(i,N) = {my(x='x+O('x^N), f=(x^i)/(1-x^i)*prod(j=i+1,N-i,1/(1-x^j))); f}
%o A188920 A_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N, B_x(i,N)*prod(j=1,i-1,1+B_x(j,N)))); Vec(f)}
%o A188920 A_x(60) \\ _John Tyler Rascoe_, Aug 23 2024
%Y A188920 For leaders of identical runs we have A000041.
%Y A188920 Matching 23-1 only gives A189076.
%Y A188920 An opposite version is A358836.
%Y A188920 For identical leaders we have A374631, ranks A374633.
%Y A188920 For distinct leaders we have A374632, ranks A374768.
%Y A188920 For weakly increasing leaders we have A374635.
%Y A188920 For non-weakly decreasing leaders we have A374636, ranks A375137.
%Y A188920 For leaders of anti-runs we have A374680.
%Y A188920 For leaders of strictly increasing runs we have A374689.
%Y A188920 The complement is counted by A375140, ranks A375295, reverse A375296.
%Y A188920 A011782 counts compositions.
%Y A188920 A238130, A238279, A333755 count compositions by number of runs.
%Y A188920 Cf. A056823, A102726, A188900, A188919, A189077, A238343, A333213, A335480/A335482, A374679, A374688.
%K A188920 nonn
%O A188920 0,3
%A A188920 _N. J. A. Sloane_, Apr 13 2011
%E A188920 More terms from _Andrew Baxter_, May 17 2011
%E A188920 a(30)-a(39) from _Alois P. Heinz_, Nov 14 2015