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 A089222 Number of ways of seating n people around a table for the second time without anyone sitting next to the same person as they did the first time. 8
 1, 0, 0, 0, 0, 10, 36, 322, 2832, 27954, 299260, 3474482, 43546872, 586722162, 8463487844, 130214368530, 2129319003680, 36889393903794, 675098760648204, 13015877566642418, 263726707757115400, 5603148830577775218, 124568968969991162100, 2892414672938546871250 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 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. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..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] Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 6 (2006), #A11. Note that in this paper a(1) = 1. See Column 2 in the table on page 3. 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) * binomial(n-m-1, k-1) * binomial(m-1, k-1) * 2^k * n * (n-m-1)!. - Joel B. 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 asymptotic 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, exact/asymptotic results are 1.0000000000125542243 (Aspvall + Liang), 1.0000000000000008990 (Kotesovec 7 terms) or 1.0000000000000000121 (Kotesovec 8 terms). - Vaclav Kotesovec, Apr 06 2012 a(n) = 2*n*A002816(n) for n > 1. - Martin Renner, Apr 01 2022 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 a(0)=1 prepended by Alois P. Heinz, Jul 31 2019 STATUS approved

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Last modified September 14 10:49 EDT 2024. Contains 375921 sequences. (Running on oeis4.)