A228708 Triangle T(n,k) read by rows: T(n,k) = number of permutations on 123...n with exactly one abc pattern and no aj pattern with j<=k, for n>=0, 0<=k<=n.

0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 6, 6, 2, 0, 0, 27, 27, 12, 3, 0, 0, 110, 110, 55, 19, 4, 0, 0, 429, 429, 229, 91, 27, 5, 0, 0, 1638, 1638, 912, 393, 136, 36, 6, 0, 0, 6188, 6188, 3549, 1614, 612, 191, 46, 7, 0, 0, 23256, 23256, 13636, 6447, 2601, 897, 257, 57, 8, 0, 0

Offset: 0

Author

N. J. A. Sloane, Sep 15 2013

Keywords

• nonn
• tabl
• easy

Comments

See Noonan-Zeilberger for precise definition.

Examples

			Triangle begins:
0
0,0
0,0,0
1,1,0,0
6,6,2,0,0
27,27,12,3,0,0
110,110,55,19,4,0,0
429,429,229,91,27,5,0,0
1638,1638,912,393,136,36,6,0,0
6188,6188,3549,1614,612,191,46,7,0,0
23256,23256,13636,6447,2601,897,257,57,8,0,0
...
		

Links

• J. Noonan and D. Zeilberger, [math/9808080] The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns. Also Adv. in Appl. Math. 17 (1996), no. 4, 381--407. MR1422065 (97j:05003).

Crossrefs

See A084249 for a curtailed version. See also A229158, A229160.

T(n, 1) = A003517(n+1). Cf. A001089.

Programs

PARI

for(n=1,15, for(k=1,n-2,print1(binomial(2*n-k-1,n)-binomial(2*n-k-1,n+3)+binomial(2*n-2*k-2,n-k-4)-binomial(2*n-2*k-2,n-k-1)+binomial(2*n-2*k-3,n-k-4)-binomial(2*n-2*k-3,n-k-2)",")))

Formula

T(n, k) = C(2n-k-1, n) - C(2n-k-1, n+3) + C(2n-2k-2, n-k-4) - C(2n-2k-2, n-k-1) + C(2n-2k-3, n-k-4) - C(2n-2k-3, n-k-2).

T(n, n-2) = n-2, T(n, k) = T(n, k+1) + T(n-1, k-1) + T(n-k, 2).