OFFSET
0,3
COMMENTS
Note that an exponentiation ^(-1) is missing in Example 4.4. The notation in Theorem 4.3 is complete.
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..300
S. Heubach, T. Mansour and A. O. Munagi, Avoiding Permutation Patterns of Type (2,1) in Compositions, Online Journal of Analytic Combinatorics, 4 (2009).
Wikipedia, Permutation pattern.
EXAMPLE
From Gus Wiseman, Aug 19 2024: (Start)
The a(6) = 31 compositions:
. (6) (5,1) (4,1,1) (3,1,1,1) (2,1,1,1,1) (1,1,1,1,1,1)
(1,5) (1,4,1) (1,3,1,1) (1,2,1,1,1)
(4,2) (1,1,4) (1,1,3,1) (1,1,2,1,1)
(2,4) (3,2,1) (1,1,1,3) (1,1,1,2,1)
(3,3) (3,1,2) (2,2,1,1) (1,1,1,1,2)
(2,3,1) (2,1,2,1)
(2,1,3) (2,1,1,2)
(1,2,3) (1,2,2,1)
(2,2,2) (1,2,1,2)
(1,1,2,2)
Missing is (1,3,2), reverse of (2,3,1).
(End)
MAPLE
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
MATHEMATICA
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}]];
a /@ Range[0, 32] (* Jean-François Alcover, Apr 02 2020, after R. J. Mathar *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MatchQ[#, {___, y_, z_, ___, x_, ___}/; x<y<z]&]], {n, 0, 15}] (* Gus Wiseman, Aug 19 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Apr 16 2011
STATUS
approved