A102726 Number of compositions of the integer n into positive parts that avoid a fixed pattern of three letters.

1, 1, 2, 4, 8, 16, 31, 60, 114, 214, 398, 732, 1334, 2410, 4321, 7688, 13590, 23869, 41686, 72405, 125144, 215286, 368778, 629156, 1069396, 1811336, 3058130, 5147484, 8639976, 14463901, 24154348, 40244877, 66911558, 111026746, 183886685, 304034456, 501877227

Offset

0,3

Comments

The sequence is the same no matter which of the six patterns of three letters is chosen as the one to be avoided.

Links

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..900 (first 400 terms from Alois P. Heinz)

M. Elder and V. Vatter, Problems and conjectures presented at the third international conference on permutation petterns

Carla D. Savage and Herbert S. Wilf, Pattern avoidance in compositions and multiset permutations, Advances in Applied Mathematics 36 (2006), pp.194-201.

Wikipedia, Permutation pattern

Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Formula

G.f.: Sum_{i>=1} (1/(1-x^i))*Product_{j>=1, j<>i} (1-x^i)/((1-x^(j-i))*(1-x^i-x^j)).

Asymptotics (Savage and Wilf, 2005): a(n) ~ c * ((1+sqrt(5))/2)^n, where c = r/(r-1)/(r-s) * (r * Product_{j>=3} (1-1/r)/(1-r^(1-j))/(1-1/r-r^(-j)) - Product_{j>=3} (1-1/r^2)/(1-r^(2-j))/(1-1/r^2-r^(-j)) ) = 18.9399867283479198666671671745270505487677312850521421513193261105... and r = (1+sqrt(5))/2, s = (1-sqrt(5))/2. - Vaclav Kotesovec, May 02 2014

Example

a(6) = 31 because there are 32 compositions of 6 into positive parts and only one of these, namely 6 = 1+2+3, contains the pattern (123), the other 31 compositions of 6 avoid that pattern.

Maple

b:= proc(n, m, t) option remember; `if`(n=0, 1,
      add(b(n-i, min(m, i, n-i), min(t, n-i,
      `if`(i>m, i, t))), i=1..min(n, t)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 18 2014

Mathematica

b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[b[n - i, Min[m, i, n - i], Min[t, n - i, If[i > m, i, t]]], {i, 1, Min[n, t]}]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 10 2017, after Alois P. Heinz *)
mstype[q_]:=q/.Table[Union[q][[i]]->i, {i, Length[Union[q]]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MemberQ[Union[mstype/@Subsets[#]], {1, 2, 3}]&]], {n, 0, 10}] (* Gus Wiseman, Jun 22 2020 *)

Prog

(PARI) seq(n)={Vec(sum(i=1, n, prod(j=1, n, if(i==j, 1, (1-x^i)/((1-x^(j-i))*(1-x^i-x^j))) + O(x*x^n))/(1-x^i)))} \\ Andrew Howroyd, Dec 31 2020

Crossrefs

The version for patterns is A226316.

These compositions are ranked by the complement of A335479.

The matching version is A335514.

The version for prime indices is A335521.

Constant patterns are counted by A000005 and ranked by A272919.

Permutations are counted by A000142 and ranked by A333218.

Patterns are counted by A000670 and ranked by A333217.

Compositions are counted by A011782.

Strict compositions are counted by A032020 and ranked by A233564.

Patterns matched by compositions are counted by A335456.

Minimal patterns avoided by a given composition are counted by A335465.

Cf. A056986, A106356, A232464, A269134, A333755.

Sequence in context: A007800 A362063 A309982 * A188900 A189075 A345372

Adjacent sequences: A102723 A102724 A102725 * A102727 A102728 A102729

Keyword

easy,nonn

Author

Herbert S. Wilf, Feb 07 2005

Extensions

More terms from Ralf Stephan, May 27 2005

Status

approved