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A089222 Number of ways of sitting n people around a table for the second time without anyone sitting next to the same person as they did the first time. 7
0, 0, 0, 0, 10, 36, 322, 2832, 27954, 299260, 3474482, 43546872, 586722162, 8463487844, 130214368530, 2129319003680, 36889393903794, 675098760648204, 13015877566642418, 263726707757115400, 5603148830577775218 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

A078603 counts these arrangements up to circular symmetry (i.e., two arrangements are the same if one can be rotated to give the other). A002816 counts them up to dihedral symmetry (i.e., two arrangements are the same if one can be rotated or reflected to give the other). [Joel B. Lewis, Jan 28 2010]

REFERENCES

J. Snell, Introduction to Probability, e-book, pp. 101 Q. 20.

Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS, vol. 6 (2006), paper A11. Note that in this paper a(1) = 1. See Column 2 in the table on page 3.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

B. Aspvall and F. M. Liang, The dinner table problem, Technical Report CS-TR-80-829, Computer Science Department, Stanford, California, 1980.

Charles M. Grinstead & J. Laurie Snell Introduction to Probability.

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 626.

Art of Problem Solving forum, Random neighbors. [From Joel B. Lewis, Jan 28 2010]

FORMULA

Inclusion-exclusion gives that for n > 2, we have a(n) = n! + 2*n*(-1)^n + sum_{1 <= k <= m < n} (-1)^m * n/k * binom(n - m - 1, k - 1) * binom(m - 1, k - 1) * 2^k * n * (n - m - 1)!  [Joel Brewster Lewis, Jan 28 2010]

a(n) = (3*n-30)*a(n-11) + (6*n-45)*a(n-10) + (5*n+18)*a(n-9) - (8*n-139)*a(n-8) - (26*n-204)*a(n-7) - (4*n-30)*a(n-6) + (26*n-148)*a(n-5) + (8*n-74)*a(n-4) - (9*n-18)*a(n-3) - (2*n-15)*a(n-2) + (n+2)*a(n-1), n>=14 [Vaclav Kotesovec, Apr 13 2010]

The asymptotic expansion from article by Aspvall and Liang (also cited in article by Tauraso) is wrong. Bad terms are 736/(15n^5)+8428/(45n^6)+40174/(63n^7)). Right asymptotics formula is a(n) ~ n!/e^2*(1 - 4/n + 20/(3n^3) + 58/(3n^4) + 796/(15n^5) + 7858/(45n^6) + 40324/(63n^7) + 140194/(63n^8) +...). Verified also numerically. For example for n=200 are exact/asymptotic results 1.0000000000125542243 (Aspvall + Liang), 1.0000000000000008990 (Kotesovec 7 terms) or 1.0000000000000000121 (Kotesovec 8 terms). [Vaclav Kotesovec, Apr 06 2012]

EXAMPLE

a(4)=0 because trying to arrange 1,2,3,4 around a table will always give a couple who is sitting next to each other and differ by 1.

MATHEMATICA

Same[cperm_, n_] := ( For[same = False; i = 2, (i <= n) && ! same, i++, same = ((Mod[cperm[[i - 1]], n] + 1) == cperm[[i]]) || ((Mod[cperm[[ i]], n] + 1) == cperm[[i - 1]])]; same = same || ((Mod[cperm[[n]], n] + 1) == cperm[[1]]) || ((Mod[ cperm[[1]], n] + 1) == cperm[[n]]); Return[same]); CntSame[n_] := (allPerms = Permutations[Range[n]]; count = 0; For[j = 1, j <= n!, j++, perm = allPerms[[j]]; If[ ! Same[perm, n], count++ ]]; Return[count]);

(* or direct computation of terms *)

Table[If[n<3, 0, n! + (-1)^n*2n + Sum[(-1)^r*(n/(n-r))^2 * (n-r)! * Sum[2^c * Binomial[r-1, c-1] * Binomial[n-r, c], {c, 1, r}], {r, 1, n-1}]], {n, 1, 25}] (* Vaclav Kotesovec, Apr 06 2012 *)

CROSSREFS

Cf. A002464, A002816, A078603, A078630, A078631.

Sequence in context: A240151 A264486 A220199 * A139242 A139236 A212795

Adjacent sequences:  A089219 A089220 A089221 * A089223 A089224 A089225

KEYWORD

nonn,nice

AUTHOR

Udi Hadad (somebody(AT)netvision.net.il), Dec 22 2003

EXTENSIONS

Tauraso reference from Parthasarathy Nambi, Dec 21 2006

More terms from Vladeta Jovovic, Nov 29 2009

STATUS

approved

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Last modified October 15 11:01 EDT 2018. Contains 316224 sequences. (Running on oeis4.)