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A123514
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Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 321 exactly once (n>=3; 1<=k<=n-2).
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1
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1, 0, 2, 4, 0, 3, 0, 10, 0, 4, 14, 0, 18, 0, 5, 0, 40, 0, 28, 0, 6, 48, 0, 81, 0, 40, 0, 7, 0, 150, 0, 140, 0, 54, 0, 8, 165, 0, 330, 0, 220, 0, 70, 0, 9, 0, 550, 0, 616, 0, 324, 0, 88, 0, 10, 572, 0, 1287, 0, 1040, 0, 455, 0, 108, 0, 11, 0, 2002, 0, 2548, 0, 1638, 0, 616, 0, 130, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 3,3
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REFERENCES
| E. Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 493 and 498).
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FORMULA
| T(n,k)=k(k+3)binomial(n+1,(n-k-2)/2)/(n+1) for n>=3, 1<=k<=n-2, n-k even.
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EXAMPLE
| T(4,2)=2 because we have 1432 and 3214 (also 4231 is an involution with 2 fixed points but contains twice the pattern 321: 421 and 431).
Triangle starts:
1;
0,2;
4,0,3;
0,10,0,4;
14,0,18,0,5;
0,40,0,28,0,6;
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MAPLE
| T:=proc(n, k) if n-k mod 2 = 0 and k<=n then k*(k+3)*binomial(n+1, (n-k)/2-1)/(n+1) else 0 fi end: for n from 3 to 15 do seq(T(n, k), k=1..n-2) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A112554, A123515.
Sequence in context: A111818 A007631 A155517 * A064178 A018220 A004580
Adjacent sequences: A123511 A123512 A123513 * A123515 A123516 A123517
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 13 2006
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