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A134435
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k odd entries that are followed by a smaller entry (n >= 1, k >= 0).
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6
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1, 2, 2, 4, 12, 12, 12, 72, 36, 144, 432, 144, 144, 1728, 2592, 576, 2880, 17280, 17280, 2880, 2880, 57600, 172800, 115200, 14400, 86400, 864000, 1728000, 864000, 86400, 86400, 2592000, 12960000, 17280000, 6480000, 518400
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OFFSET
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1,2
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COMMENTS
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Row n has ceiling(n/2) entries. T(2n,0) = T(2n+1,0) = n!(n+1)! = A010790(n).
T(n,k) is also the number of permutations of {1,2,...,n} having k adjacent pairs of the form (odd, odd) (0 <= k <= ceiling(n,2)-1). Example: T(3,1)=4 because we have 132, 213, 312 and 231. - Emeric Deutsch, Dec 14 2008
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LINKS
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Table of n, a(n) for n=1..36.
S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.
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FORMULA
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T(2n,k) = (n!)^2*C(n-1,k) C(n+1,k+1); T(2n+1,k) = n!(n+1)! * C(n,k) * C(n+1,k).
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EXAMPLE
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T(3,1) = 4 because we have 132, 312, 231 and 321.
Triangle starts:
1;
2;
2, 4;
12, 12;
12, 72, 36;
144, 432, 144;
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MAPLE
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T:=proc(n, k) if `mod`(n, 2)=0 then binomial((1/2)*n-1, k)*binomial((1/2)* n+1, k+1)*factorial((1/2)*n)^2 elif `mod`(n, 2)=1 then factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2)*binomial((1/2)*n-1/2, k)*binomial((1/2)* n+1/2, k) else 0 end if end proc: for n to 11 do seq(T(n, k), k=0..ceil((1/2)*n)-1) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A010790, A134434.
Sequence in context: A059343 A285944 A112473 * A136718 A112362 A134720
Adjacent sequences: A134432 A134433 A134434 * A134436 A134437 A134438
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch, Nov 22 2007
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STATUS
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approved
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