

A000179


Ménage numbers: number of permutations s of [0, ..., n1] such that s(i) != i and s(i) != i+1 (mod n) for all i.
(Formerly M2062 N0815)


41



1, 0, 0, 1, 2, 13, 80, 579, 4738, 43387, 439792, 4890741, 59216642, 775596313, 10927434464, 164806435783, 2649391469058, 45226435601207, 817056406224416, 15574618910994665, 312400218671253762, 6577618644576902053, 145051250421230224304, 3343382818203784146955, 80399425364623070680706, 2013619745874493923699123
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,5


COMMENTS

According to rook theory, J. Riordan considered a(1) to be 1; see A102761.  Vladimir Shevelev, Apr 02 2010
Or, for n>=3, the number of 3 X n Latin rectangles the second row of which is full cycle with a fixed order of its elements, e.g., the cycle (x_2,x_3,...,x_n,x_1) with x_1 < x_2 < ... < x_n.  Vladimir Shevelev, Mar 22 2010
Muir (p. 112) gives essentially this recurrence (although without specifying any initial conditions). Compare A186638.  N. J. A. Sloane, Feb 24 2011
Sequence discovered by Touchard in 1934.  L. Edson Jeffery, Nov 13 2013
Although these are also known as Touchard numbers, the problem was formulated by Lucas in 1891, who gave the recurrence formula shown below. See Cerasoli et al., 1988.  Stanislav Sykora, Mar 14 2014
From Vladimir Shevelev, Jun 25 2015: (Start)
According to the ménage problem, 2*n!*a(n) is the number of ways of seating n married couples at 2*n chairs around a circular table, men and women in alternate positions, so that no husband is next to his wife.
It is known [Riordan, ch. 7] that a(n) is the number of arrangements of n nonattacking rooks on the positions of the 1's in an n X n (0,1)matrix A_n with 0's in positions (i,i), i = 1,...,n, (i,i+1), i = 1,...,n1, and (n,1). This statement could be written as a(n) = per(A_n). For example, A_5 has the form
001*11
1*0011
11001* (1)
11*100
0111*0,
where 5 nonattacking rooks are denoted by {1*}.
We can indicate a onetoone correspondence between arrangements of n nonattacking rooks on the 1's of a matrix A_n and arrangements of n married couples around a circular table by the rules of the ménage problem, after the ladies w_1, w_2,..., w_n have taken the chairs numbered
2*n, 2, 4, ..., 2*n2 (2)
respectively. Suppose we consider an arrangement of rooks: (1,j_1), (2,j_2),..., (n,j_n). Then the men m_1, m_2,..., m_n took chairs with numbers
2*j_i  3 (mod 2*n), (3)
where the residues are chosen from the interval[1,2*n]. Indeed {j_i} is a permutation of 1,...,n. So {2*j_i3}(mod 2*n) is a permutation of odd positive integers <= 2*n1. Besides, the distance between m_i and w_i cannot be 1. Indeed, the equality 2*(j_ii)1 = 1 (mod 2*n) is possible if and only if either j_i=i or j_i=i+1 (mod n) that correspond to positions of 0's in matrix A_n.
For example, in the case of positions of {1*} in(1) we have j_1=3, j_2=1, j_3=5, j_4=2, j_5=4. So, by(2) and (3) the chairs 1,2,...,10 are taken by m_4, w_2, m_1, w_3, m_5, w_4, m_3, w_5, m_2, w_1, respectively. (End)
The first 20 terms of this sequence were calculated in 1891 by E. Lucas (see [Lucas, p. 495]).  Peter J. C. Moses, Jun 26 2015


REFERENCES

W. W. R. Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th Ed. Dover, p. 50.
M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Nicola Zanichelli Editore, Bologna 1988, Chapter 3, p. 78.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 185, mu(n).
Kaplansky, Irving and Riordan, John, The probleme des menages, Scripta Math. 12, (1946). 113124.
E. Lucas, Théorie des nombres, Paris, 1891, pp. 491495.
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 1, p 256.
T. Muir, A Treatise on the Theory of Determinants. Dover, NY, 1960, Sect. 132, p. 112.  N. J. A. Sloane, Feb 24 2011
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 197.
V. S. Shevelev, Reduced Latin rectangles and square matrices with equal row and column sums, Diskr. Mat. (J. of the Akademy of Sciences of Russia) 4(1992), 91110.  Vladimir Shevelev, Mar 22 2010
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff.
J. Touchard, Permutations discordant with two given permutations, Scripta Math., 19 (1953), 108119.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..100
Max Alekseyev, On enumeration of seating arrangements of couples around a circular table, Automatic Sequences Workshop, 2015
Kenneth P. Bogart and Peter G. Doyle, Nonsexist solution of the ménage problem, Amer. Math. Monthly 93 (1986), no. 7, 514519.
A. de Gennaro, How may latin rectangles are there?, arXiv:0711.0527 [math.CO] (2007), see p. 2.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 372
Nick Hobson, Python program for this sequence
Irving Kaplansky, Solution of the "Problème des ménages", Bull. Amer. Math. Soc. 49, (1943). 784785.
Irving Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906914.
S. M. Kerawala, Asymptotic solution of the "Probleme des menages, Bull. Calcutta Math. Soc., 39 (1947), 8284. [Annotated scanned copy]
V. Kotesovec, Nonattacking chess pieces, 6ed, 2013, p. 221.
A. R. Kräuter, Über die Permanente gewisser zirkulärer Matrizen...
E. Lucas, Théorie des Nombres, GauthierVillars, Paris, 1891, Vol. 1, p. 495.
H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff. [Annotated scanned copy]
J. Touchard, Théorie des substitutions. Sur un problème de permutations, C. R. Acad. Sci. Paris 198 (1934), 631633.
Eric Weisstein's World of Mathematics, Married Couples Problem
Eric Weisstein's World of Mathematics, Rooks Problem
M. Wyman and L. Moser, On the problème des ménages, Canad. J. Math., 10 (1958), 468480.
D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089, 2014


FORMULA

a(n) = ((n^22*n)*a(n1) + n*a(n2)  4(1)^n)/(n2) for n >= 4.
a(n) = A059375(n)/(2*n!).
a(n) = Sum {0<=k<=n} (1)^k*(2*n)*binomial(2*nk, k)*(nk)!/(2*nk).  Touchard (1934)
G.f.: x+(1x)/(1+x)*Sum_{n>=0} n!*(x/(1+x)^2)^n.  Vladeta Jovovic, Jun 26 2007
a(2^k+2)==0 (mod 2^k); for k>=2, a(2^k)==2(mod 2^k).  Vladimir Shevelev, Jan 14 2011
a(n) = round( 2*n*exp(2)*BesselK(n,2) ) for n>0.  Mark van Hoeij, Oct 25 2011


EXAMPLE

a(0) = 1; () works. a(1) = 0; nothing works. a(2) = 0; nothing works. a(3) = 1; (201) works. a(4) = 2; (2301), (3012) work. a(5) = 13; (20413), (23401), (24013), (24103), (30412), (30421), (34012), (34021), (34102), (40123), (43012), (43021), (43102) work.


MAPLE

A000179 := n > add ((1)^k*(2*n)*binomial(2*nk, k)*(nk)!/(2*nk), k=0..n); # for n >= 2
U := proc(n) local k; add( (2*n/(2*nk))*binomial(2*nk, k)*(nk)!*(x1)^k, k=0..n); end; W := proc(r, s) coeff( U(r), x, s ); end; A000179 := n>W(n, 0); # valid for n >= 2


MATHEMATICA

a[n_] := 2*n*Sum[(1)^k*Binomial[2*n  k, k]*(n  k)!/(2*n  k), {k, 0, n}]; a[0] = 1; a[1] = 0; Table[a[n], {n, 0, 21}] (* JeanFrançois Alcover, Dec 05 2012, from 2nd formula *)


PROG

(PARI) {a(n) = local(A); if( n<3, n==0, A = vector(n); A[3] = 1; for(k=4, n, A[k] = (k * (k  2) * A[k1] + k * A[k2]  4 * (1)^k) / (k2)); A[n])} /* Michael Somos, Jan 22 2008 */
(PARI) a(n)=if(n, round(2*n*exp(2)*besselk(n, 2)), 1) \\ Charles R Greathouse IV, Nov 03 2014
(Haskell)
import Data.List (zipWith5)
a000179 n = a000179_list !! n
a000179_list = 1 : 0 : 0 : 1 : zipWith5
(\v w x y z > (x * y + (v + 2) * z  w) `div` v) [2..] (cycle [4, 4])
(drop 4 a067998_list) (drop 3 a000179_list) (drop 2 a000179_list)
 Reinhard Zumkeller, Aug 26 2013


CROSSREFS

Diagonal of A058087. Also a diagonal of A008305.
cf. A059375, A102761, A000186, A094047, A067998, A033999.
Sequence in context: A179237 A216316 A102761 * A246383 A189087 A037739
Adjacent sequences: A000176 A000177 A000178 * A000180 A000181 A000182


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from James A. Sellers, May 02 2000
Additional comments from David W. Wilson, Feb 18 2003


STATUS

approved



