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

%I A052182 #43 Jul 02 2025 16:01:58
%S A052182 1,-3,18,-160,1875,-27216,470596,-9437184,215233605,-5500000000,
%T A052182 155624547606,-4829554409472,163086595857367,-5952860799406080,
%U A052182 233543408203125000,-9799832789158199296,437950726881001816329,-20766159817517617053696,1041273502979112415328410
%N A052182 Determinant of n X n matrix whose rows are cyclic permutations of 1..n.
%C A052182 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
%C A052182 |a(n)| = number of labeled mappings from n points to themselves (endofunctions) with an odd number of cycles. - _Vladeta Jovovic_, Mar 30 2006
%C A052182 |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
%H A052182 T. D. Noe, <a href="/A052182/b052182.txt">Table of n, a(n) for n = 1..100</a>
%H A052182 P. J. Cameron and P. Cara, <a href="http://dx.doi.org/10.1016/S0021-8693(02)00550-1">Independent generating sets and geometries for symmetric groups</a>, J. Algebra, Vol. 258, no. 2 (2002), 641-650.
%F A052182 a(n) = (-1)^(n-1) * n^(n-2) * (n^2 + n)/2.
%F A052182 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
%F A052182 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
%e A052182 a(3) = 18 because this is the determinant of [(1,2,3), (3,1,2), (2,3,1) ].
%p A052182 1,seq(LinearAlgebra:-Determinant(Matrix(n,shape=Circulant[$1..n])),n=2..30); # _Robert Israel_, Aug 31 2014
%t A052182 f[n_] := Det[ Table[ RotateLeft[ Range@ n, -j], {j, 0, n - 1}]]; Array[f, 19] (* or *)
%t A052182 f[n_] := (-1)^(n - 1)*n^(n - 2)*(n^2 + n)/2; Array[f, 19]
%t A052182 (* _Robert G. Wilson v_, Aug 31 2014 *)
%t A052182 Table[Det[Table[RotateRight[Range[k],n],{n,0,k-1}]],{k,30}] (* _Harvey P. Dale_, Jun 20 2024 *)
%o A052182 (MuPAD) (1+n)^(n-1)*binomial(n+2,n)*(-1)^(n) $ n=0..16 // _Zerinvary Lajos_, Apr 01 2007
%o A052182 (PARI) a(n) = (n+1)*(-n)^(n-1)/2; \\ _Altug Alkan_, Dec 17 2017
%Y A052182 Cf. A000312, A070896, A060281, A060435.
%K A052182 easy,sign,nice
%O A052182 1,2
%A A052182 Henry M. Gunn High School Mathematical Circle (_Joshua Zucker_), Jan 26 2000
%E A052182 More terms from _James Sellers_, Jan 31 2000