A052182 Determinant of n X n matrix whose rows are cyclic permutations of 1..n.

1, -3, 18, -160, 1875, -27216, 470596, -9437184, 215233605, -5500000000, 155624547606, -4829554409472, 163086595857367, -5952860799406080, 233543408203125000, -9799832789158199296, 437950726881001816329, -20766159817517617053696, 1041273502979112415328410

Offset: 1

Author

Henry M. Gunn High School Mathematical Circle (Joshua Zucker), Jan 26 2000

Keywords

• easy
• sign
• nice

Comments

Each row is a cyclic shift to the right by one place of the previous row. See the example below. - N. J. A. Sloane, Jan 07 2019

|a(n)| = number of labeled mappings from n points to themselves (endofunctions) with an odd number of cycles. - Vladeta Jovovic, Mar 30 2006

|a(n)| = number of functions from {1,2,...,n}->{1,2,...,n} such that of all recurrent elements the least is always mapped to the greatest. - Geoffrey Critzer, Aug 29 2013

Examples

			a(3) = 18 because this is the determinant of [(1,2,3), (3,1,2), (2,3,1) ].
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..100
• P. J. Cameron and P. Cara, Independent generating sets and geometries for symmetric groups, J. Algebra, Vol. 258, no. 2 (2002), 641-650.

Crossrefs

Cf. A000312, A070896, A060281, A060435.

Programs

Maple

1,seq(LinearAlgebra:-Determinant(Matrix(n,shape=Circulant[$1..n])),n=2..30); # Robert Israel, Aug 31 2014

Mathematica

f[n_] := Det[ Table[ RotateLeft[ Range@ n, -j], {j, 0, n - 1}]]; Array[f, 19] (* or *)
f[n_] := (-1)^(n - 1)*n^(n - 2)*(n^2 + n)/2; Array[f, 19]
(* Robert G. Wilson v, Aug 31 2014 *)
Table[Det[Table[RotateRight[Range[k],n],{n,0,k-1}]],{k,30}] (* Harvey P. Dale, Jun 20 2024 *)

MuPAD

(1+n)^(n-1)*binomial(n+2,n)*(-1)^(n) $ n=0..16 // Zerinvary Lajos, Apr 01 2007

PARI

a(n) = (n+1)*(-n)^(n-1)/2; \\ Altug Alkan, Dec 17 2017

Formula

a(n) = (-1)^(n-1) * n^(n-2) * (n^2 + n)/2.

E.g.f.[A052182] = E.g.f.[A000312] * E.g.f.[A000272], so A052182(unsigned) is "tree-like". E.g.f.: (T-T^2/2)/(1-T), where T=T(x) is Euler's tree function (see A000169). E.g.f. for signed sequence: (W+W^2/2)/(1+W), where W=W(x)=-T(-x) is the Lambert W function. - Len Smiley, Dec 13 2001

Conjecture: a(n) = -Res( f(n), x^n - 1), where Res is the resultant and f(n) = Sum_{k=1..n} k*x^k. - Benedict W. J. Irwin, Dec 07 2016

Extensions

More terms from James Sellers, Jan 31 2000