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A094112
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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 231-pattern is equal to k.
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0
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1, 0, 2, 0, 3, 3, 0, 12, 8, 4, 0, 60, 40, 15, 5, 0, 360, 240, 90, 24, 6, 0, 2520, 1680, 630, 168, 35, 7, 0, 20160, 13440, 5040, 1344, 280, 48, 8, 0, 181440, 120960, 45360, 12096, 2520, 432, 63, 9, 0, 1814400, 1209600, 453600, 120960, 25200, 4320, 630, 80, 10, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Row sums are the factorial numbers (A000142).
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REFERENCES
| E. Deutsch and W. P. Johnson, Create your own permutation statistic, Math. Mag., 77, 130-134, 2004.
R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, 383-406, 1985.
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FORMULA
| T(n, k)=n!/[(k-2)!k] for 2<=k<=n-1; T(n, n)=n; T(n, 1)=0 for n>=2; T(n, k)=0 for k>n. G.f. = sum(T(n, k)t^k z^n/n!, n, k>=1) = z[(t-1)exp(tz)+1]/(1-z).
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EXAMPLE
| T(4,3)=8 because the permutations 2134, 2143, 3124, 3142, 3241, 4123, 4132 and 4231 do not avoid all three patterns 123, 132 and 231, but their initial segments of length three, namely 213, 214, 312, 314, 324, 412, 413 and 423, do.
1; 0,2; 0,3,3; 0,12,8,4; 0,60,40,15,5; 0,360,240,90,24,6;
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MAPLE
| 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 n elif k>1 and k<n then n!/(k-2)!/k else 0 fi end: seq(seq(T(n, k), k=1..n), n=1..11);
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CROSSREFS
| Cf. A000142.
Sequence in context: A134409 A180013 A094067 * A105569 A068455 A038073
Adjacent sequences: A094109 A094110 A094111 * A094113 A094114 A094115
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004
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