login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A152660 Triangle read by rows: T(n,k) is the number of permutations of [n] for which k is the maximal number of initial entries whose parities alternate (1 <= k <= n). 1

%I

%S 1,0,2,2,2,2,8,8,0,8,48,36,12,12,12,288,216,72,72,0,72,2160,1440,576,

%T 432,144,144,144,17280,11520,4608,3456,1152,1152,0,1152,161280,100800,

%U 43200,28800,11520,8640,2880,2880,2880,1612800,1008000,432000,288000,115200,86400,28800,28800,0,28800

%N Triangle read by rows: T(n,k) is the number of permutations of [n] for which k is the maximal number of initial entries whose parities alternate (1 <= k <= n).

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

%C T(n,n) = A092186(n) (the parity alternating permutations; see the Tanimoto reference).

%C T(n,1) = A152661(n).

%H S. Tanimoto, <a href="http://arxiv.org/abs/0812.1839">Combinatorial study on the group of parity alternating permutations</a>, arXiv:0812.1839 [math.CO], 2008-2017.

%F T(2n,k) = 2(n!)^2*binomial(2n-k-1, n-floor(k/2));

%F T(2n+1,2k) = n!(n+1)!*binomial(2n-2k+1, n-k);

%F T(2n+1,2k+1) = n!(n+1)!*binomial(2n-2k, n-k-1) if k < n;

%F T(2n+1,2n+1) = n!(n+1)!.

%e T(4,2)=8 because we have 1243, 1423, 2134, 2314, 3241, 3421, 4132 and 4312.

%e Triangle starts:

%e 1;

%e 0, 2;

%e 2, 2, 2;

%e 8, 8, 0, 8;

%e 48, 36, 12, 12, 12;

%e 288, 216, 72, 72, 0, 72;

%p T := proc (n, k) if n < k then 0 elif `mod`(n, 2) = 0 and `mod`(k, 2) = 0 then 2*factorial((1/2)*n)^2*binomial(n-k-1, (1/2)*n-(1/2)*k) elif `mod`(n, 2) = 0 and `mod`(k, 2) = 1 then 2*factorial((1/2)*n)^2*binomial(n-k-1, (1/2)*n-(1/2)*k+1/2) elif `mod`(n, 2) = 1 and `mod`(k, 2) = 0 then factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2)*binomial(n-k, (1/2)*n-(1/2)*k-1/2) elif `mod`(n, 2) = 1 and k = n then factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2) else factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2)*binomial(n-k, (1/2)*n-(1/2)*k-1) end if end proc: for n to 10 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

%t T[n0_?EvenQ, k_] := With[{n = n0/2}, 2 (n!)^2*Binomial[2 n - k - 1, n - Floor[k/2]]];

%t T[n1_?OddQ, k0_?EvenQ] := With[{n = (n1 - 1)/2, k = k0/2}, n! (n + 1)! * Binomial[2 n - 2 k + 1, n - k] ];

%t T[n1_?OddQ, k1_?OddQ] := With[{n = (n1 - 1)/2, k = (k1 - 1)/2}, n! (n+1)! * Binomial[2 n - 2 k, n - k - 1] ];

%t T[n1_?OddQ, n1_?OddQ] := With[{n = (n1 - 1)/2}, n! (n + 1)!];

%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-Fran├žois Alcover_, Nov 28 2017 *)

%Y Cf. A000142, A092186, A152661.

%K nonn,tabl

%O 1,3

%A _Emeric Deutsch_, Dec 12 2008

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 09:25 EST 2019. Contains 329791 sequences. (Running on oeis4.)