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 A189076 #39 Aug 22 2024 09:18:21
%S A189076 1,1,2,4,8,16,31,61,118,228,440,846,1623,3111,5955,11385,21752,41530,
%T A189076 79250,151161,288224,549408,1047034,1995000,3800662,7239710,13789219,
%U A189076 26261678,50012275,95237360,181350695,345315255,657506300,1251912618,2383636280,4538364446
%N A189076 Number of compositions of n that avoid the pattern 23-1.
%C A189076 Note that an exponentiation ^(-1) is missing in Example 4.4. The notation in Theorem 4.3 is complete.
%C A189076 Theorem: The reverse of a composition avoids 23-1 iff its leaders of maximal weakly increasing runs are weakly decreasing. For example, the composition y = (3,2,1,2,2,1,2,5,1,1,1) has maximal weakly increasing runs ((3),(2),(1,2,2),(1,2,5),(1,1,1)), with leaders (3,2,1,1,1), which are weakly decreasing, so the reverse of y is counted under a(21). - _Gus Wiseman_, Aug 19 2024
%H A189076 Alois P. Heinz, <a href="/A189076/b189076.txt">Table of n, a(n) for n = 0..300</a>
%H A189076 S. Heubach, T. Mansour and A. O. Munagi, <a href="https://web.math.rochester.edu/misc/ojac/vol4/CompType21.pdf">Avoiding Permutation Patterns of Type (2,1) in Compositions</a>, Online Journal of Analytic Combinatorics, 4 (2009).
%H A189076 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>.
%H A189076 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A189076 From _Gus Wiseman_, Aug 19 2024: (Start)
%e A189076 The a(6) = 31 compositions:
%e A189076 . (6) (5,1) (4,1,1) (3,1,1,1) (2,1,1,1,1) (1,1,1,1,1,1)
%e A189076 (1,5) (1,4,1) (1,3,1,1) (1,2,1,1,1)
%e A189076 (4,2) (1,1,4) (1,1,3,1) (1,1,2,1,1)
%e A189076 (2,4) (3,2,1) (1,1,1,3) (1,1,1,2,1)
%e A189076 (3,3) (3,1,2) (2,2,1,1) (1,1,1,1,2)
%e A189076 (2,3,1) (2,1,2,1)
%e A189076 (2,1,3) (2,1,1,2)
%e A189076 (1,2,3) (1,2,2,1)
%e A189076 (2,2,2) (1,2,1,2)
%e A189076 (1,1,2,2)
%e A189076 Missing is (1,3,2), reverse of (2,3,1).
%e A189076 (End)
%p A189076 A189075 := proc(n) local g,i; g := 1; for i from 1 to n do 1-x^i/mul ( 1-x^j,j=i+1..n-i) ; g := g*% ; end do: g := expand(1/g) ; g := taylor(g,x=0,n+1) ; coeftayl(g,x=0,n) ; end proc: # _R. J. Mathar_, Apr 16 2011
%t A189076 a[n_] := Module[{g = 1, xi}, Do[xi = 1 - x^i/Product[1 - x^j, {j, i+1, n-i}]; g = g xi, {i, n}]; SeriesCoefficient[1/g, {x, 0, n}]];
%t A189076 a /@ Range[0, 32] (* _Jean-François Alcover_, Apr 02 2020, after _R. J. Mathar_ *)
%t A189076 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{___,y_,z_,___,x_,___}/;x<y<z]&]],{n,0,15}] (* _Gus Wiseman_, Aug 19 2024 *)
%Y A189076 The non-dashed version is A102726.
%Y A189076 The version for 3-12 is A188900, complement A375406.
%Y A189076 Avoiding 12-1 also gives A188920 in reverse.
%Y A189076 The version for 13-2 is A189077.
%Y A189076 For identical leaders we have A374631, ranks A374633.
%Y A189076 For distinct leaders we have A374632, ranks A374768.
%Y A189076 The complement is counted by A374636, ranks A375137.
%Y A189076 A011782 counts compositions.
%Y A189076 A238130, A238279, A333755 count compositions by number of runs.
%Y A189076 Cf. A056823, A188919, A238343, A274174, A333213, A335480, A358836, A374635.
%K A189076 nonn
%O A189076 0,3
%A A189076 _N. J. A. Sloane_, Apr 16 2011