A094067 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding the 123-, the 132- and the 321-pattern is equal to k.

1, 0, 2, 0, 3, 3, 0, 12, 7, 5, 0, 60, 35, 17, 8, 0, 360, 210, 102, 35, 13, 0, 2520, 1470, 714, 245, 70, 21, 0, 20160, 11760, 5712, 1960, 560, 134, 34, 0, 181440, 105840, 51408, 17640, 5040, 1206, 251, 55, 0, 1814400, 1058400, 514080, 176400, 50400, 12060

Offset

1,3

Comments

Row sums are the factorial numbers (A000142).

Diagonal yields the Fibonacci numbers A000045.

Links

Table of n, a(n) for n=1..52.

E. Deutsch and W. P. Johnson, Create your own permutation statistics, Math. Mag., 77, 130-134, 2004.

R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, 383-406, 1985.

Formula

T(n, k) = n!*[(k+1)fibonacci(k+1)-fibonacci(k+2)]/(k+1)! for 1<=k<=n-1; T(1, 1)=1; T(n, n)=fibonacci(n+1).

Example

T(4,3)=7 because the permutations 4132, 3124, 2413, 4213, 2314 and 3214 do not avoid all three patterns 123, 132 and 213, but their initial segments of length three, namely 413, 312, 241, 421, 231 and 321, do.
Triangle begins:
1;
0,2;
0,3,3;
0,12,7,5;
0,60,35,17,8;
0,360,210,102,35,13;
0,2520,1470,714,245,70,21;

Maple

with(combinat): T:=proc(n, k) if n=1 and k=1 then 1 elif n=1 then 0 elif k=1 then 0 elif k=n then fibonacci(n+1) elif k>0 and k<n then n!*((k+1)*fibonacci(k+1)-fibonacci(k+2))/(k+1)! else 0 fi end: seq(seq(T(n, k), k=1..n), n=1..11);

Crossrefs

Cf. A000142, A000045.

Sequence in context: A298605 A180013 A377657 * A094112 A333303 A326926

Adjacent sequences: A094064 A094065 A094066 * A094068 A094069 A094070

Keyword

nonn,tabl

Author

Emeric Deutsch, May 31 2004

Status

approved