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A098916 Permanent of the n X n (0,1)-matrices with ij-th entry equal to zero iff (i=1,j=1),(i=1,j=n),(i=n,j=1) and (i=n,j=n). 5
0, 4, 36, 288, 2400, 21600, 211680, 2257920, 26127360, 326592000, 4390848000, 63228211200, 971415244800, 15866448998400, 274611617280000, 5021469573120000, 96746980442112000, 1959126353952768000 (list; graph; refs; listen; history; text; internal format)



The number of all possible ways to permute n distinct aligned balls, one is blue, 2 are red and the remaining are green, such that no red ball occurs by the side of the blue ball. It may generalized to r red balls: a(n,r) = (n-r-1)(n-r)(n-2)!. - Alessandro Nicolosi (xxalenicxx(AT)hotmail.com), Jul 12 2006

A formula for the permanents of these n X n matrices(A) can be easily derived by minor expansion along the first row: a(n)=per(A)=(n-2)*per(B), where B is the n-1 X n-1 (0,1)-matrix with bij=0 iff (i=n,j=1) and (i=n,j=n). A new minor expansion along the last row of B yields: per(B)=(n-3)*per(C)=(n-3)*(n-2)! since C is the n-2 X n-2 1-matrix. Hence: a(n)=(n-2)*(n-3)*(n-2)!. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007

Number of permutations of n-1 having exactly 4 points P on the boundary of their bounding square. (A bounding square for a permutation of n is the square with sides parallel to the coordinate axis containing (1,1) and (n,n), and the set of points P of a permutation p is the set {(k,p(k)) for 0<k<n+1}). - David Nacin, Feb 27 2012

a(n) is also the number of permutations of n symbols that 4-commute with a transposition (see A233440 for definition): a permutation p of {1,...,n} has exactly four points on the boundary of their bounding square if and only if p 4-commutes with transposition (1, n). - Luis Manuel Rivera Martínez, Feb 27 2014


Vincenzo Librandi, Table of n, a(n) for n = 3..200

Emeric Deutsch, Permutations and their bounding squares, Math Magazine, 85(1) (2012), p. 63.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.


a(n) = (n-2)*(n-3)*(n-2)!. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007

From Amiram Eldar, May 17 2022: (Start)

Sum_{n>=4} 1/a(n) = 3 - e, where e = A001113.

Sum_{n>=4} (-1)^n/a(n) = 2*(gamma - Ei(-1)) - 1/e - 1, where gamma = A001620 and Ei(-1) = -A099285. (End)


a(3) = 0 because no configuration is allowed, the 2 red balls always occurs by the side of the blue ball. a(4) = 4 because we can have 4 possible permutations: b,g1,r1,r2 b,g1,r2,r1 r1,r2,g1,b r2,r1,g1,b.


a:= n->(n-2)*(n-3)*(n-2)!: seq(a(n), n=3..20); # Zerinvary Lajos, Jul 01 2007


a[n_, r_] := (n-r-1)(n-r)(n-2)! (* Alessandro Nicolosi (xxalenicxx(AT)hotmail.com), Jul 12 2006 *)

Table[(n-2)*(n-3)*(n-2)!, {n, 3, 30}] (* Vincenzo Librandi, Feb 27 2012 *)


(PARI) permRWNb(a)=n=matsize(a)[1]; if(n==1, return(a[1, 1])); sg=1; in=vectorv(n); x=in; x=a[, n]-sum(j=1, n, a[, j])/2; p=prod(i=1, n, x[i]); for(k=1, 2^(n-1)-1, sg=-sg; j=valuation(k, 2)+1; z=1-2*in[j]; in[j]+=z; x+=z*a[, j]; p+=prod(i=1, n, x[i], sg)); return(2*(2*(n%2)-1)*p)

for(n=3, 24, a=matrix(n, n, i, j, 1); a[1, 1]=0; a[1, n]=0; a[n, 1]=0; a[n, n]=0; print1(permRWNb(a)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007

(PARI) for(n=3, 24, print1((n-2)*(n-3)*(n-2)!", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007


import math

def a(n):

return (n-2)*(n-3)*math.factorial(n-2) # David Nacin, Feb 27 2012


Cf. A208528, A208529.

Cf. A001113, A001620, A099285.

Sequence in context: A345633 A173429 A172134 * A354401 A354404 A316297

Adjacent sequences: A098913 A098914 A098915 * A098917 A098918 A098919




Simone Severini, Oct 17 2004


More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007



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