A077612 -id:A077612 - cp's OEIS frontend

A077613 Number of adjacent pairs of form (even,odd) among all permutations of {1,2,...,n}. Also, number of adjacent pairs of form (odd,even).

0, 1, 4, 24, 144, 1080, 8640, 80640, 806400, 9072000, 108864000, 1437004800, 20118067200, 305124019200, 4881984307200, 83691159552000, 1506440871936000, 28810681675776000, 576213633515520000, 12164510040883200000, 267619220899430400000, 6182004002776842240000

Offset: 1

Leroy Quet, Frank Ruskey and Dean Hickerson, Nov 11 2002

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..449

Cf. A000142, A002620, A077611, A077612.

Cf. A001620, A068985, A099284.

Array[Floor[#/2] Ceiling[#/2] (# - 1)! &, 19] (* Michael De Vlieger, Aug 16 2017 *)

a(n) = floor(n/2)*ceil(n/2)*(n-1)!; \\ Michel Marcus, Aug 29 2013

a(n) = floor(n/2)*ceiling(n/2)*(n-1)!. Proof: There are floor(n/2)*ceiling(n/2) pairs (r, s) with r even and s odd. For each pair, there are n-1 places it can occur in a permutation and (n-2)! possible arrangements of the other numbers.
a(n) = A002620(n) * A000142(n-1). - Michel Marcus, Aug 29 2013
Sum_{n>=2} 1/a(n) = 6*(CoshIntegral(1) - gamma) + 2/e - 1 = 6*(A099284 - A001620) + 2*A068985 - 1. - Amiram Eldar, Jan 22 2023

A077611 Number of adjacent pairs of form (odd,odd) among all permutations of {1,2,...,n}.

Original entry on oeis.org

0, 0, 4, 12, 144, 720, 8640, 60480, 806400, 7257600, 108864000, 1197504000, 20118067200, 261534873600, 4881984307200, 73229764608000, 1506440871936000, 25609494822912000, 576213633515520000, 10948059036794880000, 267619220899430400000, 5620003638888038400000

Offset: 1

Leroy Quet, Frank Ruskey, and Dean Hickerson, Nov 11 2002

nonn

a(n) is also the number of permutations of [n+1] starting and ending with an even number. - Olivier Gérard, Nov 07 2011

			For n=4, the a(4) = 12 permutations of degree 5 starting and ending with an even number are 21354, 21534, 23154, 23514, 25134, 25314, 41352, 41532, 43152, 43512, 45132, 45312.

Vincenzo Librandi, Table of n, a(n) for n = 1..400

Cf. A052618, A077612, A077613.

Cf. A001620, A073742, A099284.

[Factorial(n-1)*(2*n*(n-1)-(2*n-1)*(-1)^n-1)/8 : n in [1..30]]; // Vincenzo Librandi, Nov 16 2011

Table[Ceiling[n/2] Ceiling[n/2 - 1] (n - 1)!, {n, 22}] (* Michael De Vlieger, Aug 20 2017 *)

a(n) = ceiling(n/2)*ceiling(n/2-1)*(n-1)!. Proof: There are ceiling(n/2) * ceiling(n/2-1) pairs (r, s) with r and s odd and distinct. For each pair, there are n-1 places it can occur in a permutation and (n-2)! possible arrangements of the other numbers.
a(n) = (n-1)!*(2*n*(n-1)-(2*n-1)*(-1)^n-1)/8. - Bruno Berselli, Nov 07 2011
Sum_{n>=3} 1/a(n) = 4*(CoshIntegral(1) - gamma - sinh(1) + 1) = 4*(A099284 - A001620 - A073742 + 1). - Amiram Eldar, Jan 22 2023

A145892 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k adjacent pairs of the form (even,even) (0<=k<=floor(n/2)-1).

Original entry on oeis.org

1, 1, 2, 6, 12, 12, 72, 48, 144, 432, 144, 1440, 2880, 720, 2880, 17280, 17280, 2880, 43200, 172800, 129600, 17280, 86400, 864000, 1728000, 864000, 86400, 1814400, 12096000, 18144000, 7257600, 604800, 3628800, 54432000, 181440000, 181440000, 54432000, 3628800

Offset: 0

Emeric Deutsch, Nov 30 2008

Row n contains floor(n/2) entries (n>=2).
Sum of entries in row n = n! = A000142(n).
Sum_{k>=0} k*T(n,k) = A077612(n).
T(2n,k) = A134435(2n,k).

			T(4,1) = 12 because we have 1243, 1423, 1324, 1342, 3124, 3142, 2413, 4213, 2431, 4231, 3241 and 3421.
Triangle starts:
     1;
     1;
     2;
     6;
    12,   12;
    72,   48;
   144,  432, 144;
  1440, 2880, 720;
  ...

Cf. A000142, A077612, A134434, A134435, A145891.

T:=proc(n,k) if `mod`(n, 2) = 0 then factorial((1/2)*n)^2*binomial((1/2)*n-1, k)*binomial((1/2)*n+1, k+1) else factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2)*binomial((1/2)*n-3/2, k)*binomial((1/2)* n+3/2, k+2) end if end proc: 1; 1; for n from 2 to 12 do seq(T(n, k), k = 0 .. floor((1/2)*n)-1) end do; # yields sequence in triangular form

T[n_,k_]:=If[EvenQ[n],((n/2)!)^2Binomial[n/2-1,k]Binomial[n/2+1,k+1], ((n-1)/2)!((n+1)/2)!Binomial[(n-3)/2,k]Binomial[(n+3)/2,k+2]]; Join[{1,1},Flatten[Table[T[n,k],{n,0,12},{k,0,Floor[n/2]-1}]]] (* Stefano Spezia, Jul 12 2024 *)

T(2n,k) = (n!)^2*C(n-1,k)*C(n+1,k+1); T(2n+1,k) = n!(n+1)! * C(n-1,k) * C(n+2,k+2).

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A077613 Number of adjacent pairs of form (even,odd) among all permutations of {1,2,...,n}. Also, number of adjacent pairs of form (odd,even).

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A077611 Number of adjacent pairs of form (odd,odd) among all permutations of {1,2,...,n}.

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A145892 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k adjacent pairs of the form (even,even) (0<=k<=floor(n/2)-1).

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