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A152874
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} with k parity changes (n>=2; 1<=k <=n-1; the permutation 372185946 has 5 parity changes: 37-2-1-8-59-46.
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1
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2, 4, 2, 8, 8, 8, 24, 36, 48, 12, 72, 144, 288, 144, 72, 288, 720, 1728, 1296, 864, 144, 1152, 3456, 10368, 10368, 10368, 3456, 1152, 5760, 20160, 69120, 86400, 103680, 51840, 23040, 2880, 28800, 115200, 460800, 691200, 1036800, 691200, 460800
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 2,1
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COMMENTS
| Sum of entries in row n is n! (A000142(n)).
T(n,n-1)=A092186(n).
T(n,1)=A152875(n).
Sum(k*T(n,k),k=1..n-1)=2*A077613(n).
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FORMULA
| T(2n,k)=(n!)^2*a(n,k), where a(n,k) is the number of lattice paths from (0,0) to (n,n) with steps N=(0,1) and E=(1,0) and having k turns;
a(n,k)=2*binom(n-1,k/2-1)*binom(n-1,k/2) if k even;
a(n,k)=2*[binom(n-1,(k-1)/2)]^2 if k odd.
T(2n+1,k)=n!*(n+1)!*b(n,k), where b(n,k) is the number of lattice paths from (0,0) to (n,n+1) with steps N=(0,1) and E=(1,0) and having k turns;
b(n,k)=binom(n,k/2)*binom(n-1,k/2-1)+binom(n,k/2-1)*binom(n-1,k/2)=[binom(n,k/2)]^2*k(2n-k+1)/[n(2n-k+2)] if k even;
b(n,k)=2*binom(n,(k-1)/2)*binom(n-1,(k-1)/2) if k odd.
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EXAMPLE
| T(4,3)=8 because we have 1243, 1423, 4132, 4312, 2134, 2314, 3241 and 3421.
Triangle starts:
2;
4,2;
8,8,8;
24,36,48,12;
72,144,288,144,72
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MAPLE
| ae := proc (n, k) if `mod`(k, 2) = 0 then 2*factorial(n)^2*binomial(n-1, (1/2)*k-1)*binomial(n-1, (1/2)*k) else 2*factorial(n)^2*binomial(n-1, (1/2)*k-1/2)^2 end if end proc: ao := proc (n, k) if `mod`(k, 2) = 0 then factorial(n)*factorial(n+1)*(binomial(n, (1/2)*k)*binomial(n-1, (1/2)*k-1)+binomial(n, (1/2)*k-1)*binomial(n-1, (1/2)*k)) else 2*factorial(n)*factorial(n+1)*binomial(n, (1/2)*k-1/2)*binomial(n-1, (1/2)*k-1/2) end if end proc: T := proc (n, k) if `mod`(n, 2) = 0 then ae((1/2)*n, k) else ao((1/2)*n-1/2, k) end if end proc: for n from 2 to 10 do seq(T(n, k), k = 1 .. n-1) end do; # yields sequence in triangular form
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CROSSREFS
| A000142, A092186, A077613
Sequence in context: A204898 A113477 A129178 * A065286 A068217 A114593
Adjacent sequences: A152871 A152872 A152873 * A152875 A152876 A152877
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2008
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