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A152874 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. 1
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; text; internal format)
OFFSET

2,1

COMMENTS

Sum of entries in row n is n! (A000142(n)).

T(n,n-1) = A092186(n).

T(n,1) = A152875(n).

Sum_{k=1..n-1} k*T(n,k) = 2*A077613(n).

LINKS

Table of n, a(n) for n=2..44.

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*binomial(n-1,k/2-1)*binomial(n-1,k/2) if k even;

a(n,k) = 2*(binomial(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) = binomial(n,k/2)*binomial(n-1,k/2-1) + binomial(n,k/2-1)*binomial(n-1,k/2) = (binomial(n,k/2))^2*k(2n-k+1)/(n(2n-k+2)) if k even;

b(n,k) = 2*binomial(n,(k-1)/2)*binomial(n-1,(k-1)/2) if k odd.

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;

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

CROSSREFS

Cf. A000142, A077613, A092186.

Sequence in context: A336093 A304213 A129178 * A324716 A328378 A296429

Adjacent sequences:  A152871 A152872 A152873 * A152875 A152876 A152877

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Dec 15 2008

STATUS

approved

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Last modified April 14 07:59 EDT 2021. Contains 342946 sequences. (Running on oeis4.)