A155518 -id:A155518

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A155517

Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} for which the number of j < ceiling(n/2) such that p(j) + p(n+1-j) = n+1 is equal to k (n>=1; 0<=k <=ceiling(n/2)).

+10
2

0, 1, 0, 2, 4, 0, 2, 16, 0, 8, 64, 48, 0, 8, 384, 288, 0, 48, 2880, 1536, 576, 0, 48, 23040, 12288, 4608, 0, 384, 208896, 115200, 30720, 7680, 0, 384, 2088960, 1152000, 307200, 76800, 0, 3840, 23193600, 12533760, 3456000, 614400, 115200, 0, 3840, 278323200

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

For the permutation 31756284 of S_8 we have k=2 because p(2) + p(7) = 1+8 = 9 and p(3) + p(6) = 7+2 = 9; for the permutation 3214756 of S_7 we have k=2 because p(3) + p(5) = 1+7 = 8 and p(4) + p(4) = 4+4 = 8.

Row sums are the factorial numbers (A000142).

Row n contains 1 + ceiling(n/2)entries.

T(2n,n) = n!*2^n = A037223(2n) = number of centrosymmetric permutations in S[2n];

T(2n+1,n+1) = n!*2^n = A037223(2n+1) = number of centrosymmetric permutations in S[2n+1].

T(n,0) = A155518(n).

Sum_{k=0..ceiling(n/2)} k*T(n,k) = A155519(n).

LINKS

Table of n, a(n) for n=1..48.

FORMULA

T(2n,k) = n!*2^n*A055140(n,k);

T(2n-1,k) = (n-1)!*2^(n-1)*A055140(n,k);

here A055140(n,k) = A053871(n-k)*binomial(n,k), where g(n) = A053871(n) is defined by g(0)=1, g(1)=0, g(n) = 2(n-1)(g(n-1)+g(n-2)).

EXAMPLE

T(4,2)=8 because we have 1234, 4231, 1324, 4321, 2143, 3142, 2413 and 3412.

Triangle starts:

0, 1;

0, 2;

4, 0, 2;

16, 0, 8;

64, 48, 0, 8;

384, 288, 0, 48;

MAPLE

g[0] := 1: g[1] := 0: for n from 2 to 20 do g[n] := (2*(n-1))*(g[n-1]+g[n-2]) end do: T := proc (n, k) if `mod`(n, 2) = 0 then 2^((1/2)*n)*factorial((1/2)*n)*g[(1/2)*n-k]*binomial((1/2)*n, k) else 2^((1/2)*n-1/2)*factorial((1/2)*n-1/2)*g[(1/2)*n+1/2-k]*binomial((1/2)*n+1/2, k) end if end proc: for n to 12 do seq(T(n, k), k = 0 .. ceil((1/2)*n)) end do;

CROSSREFS

Cf. A000142, A037223, A055140, A155518, A155519.

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jan 26 2009

STATUS

approved

A155519

a(n) = Sum (J(p): p is a permutation of {1,2,...,n}), where J(p) is the number of j <= ceiling(n/2) such that p(j) + p(n+1-j) = n+1.

+10
2

1, 2, 4, 16, 72, 432, 2880, 23040, 201600, 2016000, 21772800, 261273600, 3353011200, 46942156800, 697426329600, 11158821273600, 188305108992000, 3389491961856000, 64023737057280000, 1280474741145600000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(n) = Sum_{k=0..ceiling(n/2)} k*A155517(n,k).

LINKS

Table of n, a(n) for n=1..20.

FORMULA

a(2n-1) = n(2n-2)!; a(2n) = 2(2n-2)!*n^2.

EXAMPLE

a(3)=4 because J(123)=2 (counting j=1,2), J(321)=2 (counting j=1,2) and J(132) = J(312) = J(213) = J(231) = 0.

MAPLE

a := proc (n) if `mod`(n, 2) = 1 then (1/2)*(n+1)*factorial(n-1) else (1/2)*factorial(n-2)*n^2 end if end proc: seq(a(n), n = 1 .. 23);