A228708 -id:A228708 - cp's OEIS frontend

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

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

Ralf Stephan, May 21 2003

See A228708 for further information.

			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
...

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).

See A228708 for the full triangle.

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

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)",")))

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).

A229158 Triangle T(n,k) read by rows: T(n,k) = number of permutations on 123...n with exactly two abc patterns and no aj pattern with j<=k, for n>=0, 0<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 1, 0, 0, 24, 24, 12, 2, 0, 0, 133, 133, 74, 23, 3, 0, 0, 635, 635, 371, 141, 36, 4, 0, 0, 2807, 2807, 1688, 709, 227, 51, 5, 0, 0, 11864, 11864, 7276, 3248, 1168, 334, 68, 6, 0, 0, 48756, 48756, 30340, 14121, 5459, 1771, 464, 87, 7, 0, 0

Offset: 0

N. J. A. Sloane, Sep 15 2013

See Noonan-Zeilberger for precise definition.

			Triangle begins:
0,
0,0,
0,0,0,
0,0,0,0,
3,3,1,0,0,
24,24,12,2,0,0,
133,133,74,23,3,0,0,
635,635,371,141,36,4,0,0,
2807,2807,1688,709,227,51,5,0,0,
11864,11864,7276,3248,1168,334,68,6,0,0,
48756,48756,30340,14121,5459,1771,464,87,7,0,0
...

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).

Cf. A228708, A001089.

A229160 Triangle T(n,k) read by rows: T(n,k) = number of permutations on 123...n with exactly one cab pattern and no aj pattern with j<=k, for n>=0, 0<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 5, 5, 2, 0, 0, 21, 21, 11, 3, 0, 0, 84, 84, 49, 19, 4, 0, 0, 330, 330, 204, 92, 29, 5, 0, 0, 1287, 1287, 825, 405, 153, 41, 6, 0, 0, 5005, 5005, 3289, 1705, 715, 235, 55, 7, 0, 0, 19448, 19448, 13013, 7007, 3146, 1166, 341, 71, 8, 0, 0

Offset: 0

N. J. A. Sloane, Sep 15 2013

See Noonan-Zeilberger for precise definition.

			Triangle begins:
0,
0,0,
0,0,0,
1,1,0,0,
5,5,2,0,0,
21,21,11,3,0,0,
84,84,49,19,4,0,0,
330,330,204,92,29,5,0,0,
1287,1287,825,405,153,41,6,0,0,
5005,5005,3289,1705,715,235,55,7,0,0,
19448,19448,13013,7007,3146,1166,341,71,8,0,0,
...

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).

Cf. A228708, A229158, A002054.

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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

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A229158 Triangle T(n,k) read by rows: T(n,k) = number of permutations on 123...n with exactly two abc patterns and no aj pattern with j<=k, for n>=0, 0<=k<=n.

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A229160 Triangle T(n,k) read by rows: T(n,k) = number of permutations on 123...n with exactly one cab pattern and no aj pattern with j<=k, for n>=0, 0<=k<=n.

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Author

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