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Search: id:A123515
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| A123515 |
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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 231 exactly once (n>=4, 2<=k<=n-2). |
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+0
2
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| 1, 0, 2, 2, 0, 3, 0, 8, 0, 4, 5, 0, 18, 0, 5, 0, 26, 0, 32, 0, 6, 12, 0, 75, 0, 50, 0, 7, 0, 76, 0, 164, 0, 72, 0, 8, 28, 0, 264, 0, 305, 0, 98, 0, 9, 0, 208, 0, 680, 0, 510, 0, 128, 0, 10, 64, 0, 840, 0, 1460, 0, 791, 0, 162, 0, 11, 0, 544, 0, 2480, 0, 2772, 0, 1160, 0, 200, 0, 12, 144
(list; table; graph; listen)
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OFFSET
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4,3
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COMMENT
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Also the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 312 exactly once (n>=4, 2<=k<=n-2). Example: T(5,3)=2 because we have 15342 and 42315 (also the involution 52341 has 3 fixed points but it contains 3 times the pattern 312: 523, 524, and 534).
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REFERENCES
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E. Deutsch, A. Robertson, and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 492 and 498).
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FORMULA
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T(n,k)=2^((n-k-6)/2)*(k-1)[binomial((n+k)/2-2, (n-k)/2-1)+2binomial((n+k)/2-3, (n-k)/2-1)+binomial((n+k)/2-4, (n-k)/2-1)] for n>=4, n+k even; T(n,k)=0 otherwise.
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EXAMPLE
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T(5,3)=2 because we have 15342 and 42315 (also the involution 52341 has 3 fixed points but it contains 3 times the pattern 231: 231, 241, and 341).
Triangle starts:
1;
0,2;
2,0,3;
0,8,0,4;
5,0,18,0,5;
0,26,0,32,0,6;
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MAPLE
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T:=proc(n, k) if n>=4 and n+k mod 2 = 0 then (k-1)*2^((n-k-6)/2)*(binomial((n+k)/2-2, (n-k)/2-1)+2*binomial((n+k)/2-3, (n-k)/2-1)+binomial((n+k)/2-4, (n-k)/2-1)) else 0 fi end: for n from 4 to 16 do seq(T(n, k), k=2..n-2) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A123514, A112554.
Sequence in context: A087319 A101348 A141659 this_sequence A058648 A112174 A089990
Adjacent sequences: A123512 A123513 A123514 this_sequence A123516 A123517 A123518
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 13 2006
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