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A084249
Triangle T(n,k) read by rows: permutations on 123...n with one abc pattern and no aj pattern with j<=k, n>2, k<n-1.
3
1, 6, 2, 27, 12, 3, 110, 55, 19, 4, 429, 229, 91, 27, 5, 1638, 912, 393, 136, 36, 6, 6188, 3549, 1614, 612, 191, 46, 7, 23256, 13636, 6447, 2601, 897, 257, 57, 8, 87210, 52020, 25332, 10695, 3951, 1260, 335, 69, 9, 326876, 197676, 98532
OFFSET
3,2
COMMENTS
See A228708 for further information.
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).
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).
EXAMPLE
Full 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
...
PROG
(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)", ")))
CROSSREFS
See A228708 for the full triangle.
T(n, 1) = A003517(n+1). Cf. A001089.
Sequence in context: A142707 A305874 A176965 * A176591 A381559 A362989
KEYWORD
nonn,tabl,easy
AUTHOR
Ralf Stephan, May 21 2003
STATUS
approved