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 A272641 Number of permutations of [1..n] that achieve a lower bound on the dominating set. 2
 1, 2, 2, 24, 64, 80, 3408, 9856, 13440, 1377792, 4139520, 5913600, 1191370752, 3659335680, 5381376000, 1878991994880, 5854937088000, 8782405632000, 4877246236262400, 15351645044736000, 23361198981120000, 19383120526049280000, 61467986759122944000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Michael De Vlieger, Table of n, a(n) for n = 1..477 C. Coscia, J. DeWitt, F. Yang, Y. Zhang, Online and Random Domination of Graphs, arXiv preprint arXiv:1509.08876 [math.CO], 2015. Jonathan Dewitt, Christopher Coscia, Fan Yang, Yiguang Zhang, Best and Worst Case Permutations for Random Online Domination of the Path, Discrete Mathematics & Theoretical Computer Science, December 20, 2017, Vol. 19 no. 2, Permutation Patterns 2016. FORMULA Section 4 of Coscia et al. 2015 gives formulas. MAPLE A272641 := proc(n) if modp(n, 3) = 0 then n!/3^(n/3) ; elif modp(n, 3) = 1 then if n > 7 then 720*binomial(n, 7)*(n-4)/3*(n-7)!/3^((n-7)/3) +2*(24*binomial(n, 2)*binomial(n-2, 5)*(n-7)!/3^((n-7)/3) *(n-4)/3 +9*binomial(n, 4) *(n-4)! /3^((n-4)/3) ) +6*binomial(n, 4)*(n-4)!/3^((n-4)/3) +24^2 *binomial(10, 5) *binomial(n, 10) *binomial((n-4)/3, 2) *(n-10)! /3^((n-10)/3) ; elif n = 1 then 1; elif n = 4 then 24; elif n = 7 then 3408; end if; else if n > 2 then 24*binomial(n, 5) *(n-5)! /3^((n-5)/3) *(n-2)/3 +2*binomial(n, 2) *(n-2)! /3^((n-2)/3) ; else 2; end if; end if ; end proc: # R. J. Mathar, May 11 2016 MATHEMATICA a[n_] := Which[Mod[n, 3] == 0, n!/3^(n/3), Mod[n, 3] == 1, Which[n > 7, 720*Binomial[n, 7]*(n - 4)/3*(n - 7)!/3^((n - 7)/3) + 2*(24*Binomial[n, 2]*Binomial[n - 2, 5]*(n - 7)!/3^((n - 7)/3)*(n - 4)/3 + 9*Binomial[n, 4] *(n - 4)! /3^((n - 4)/3)) + 6*Binomial[n, 4]*(n - 4)!/3^((n - 4)/3) + 24^2 *Binomial[10, 5]*Binomial[n, 10]*Binomial[(n - 4)/3, 2]*(n - 10)! /3^((n - 10)/3), n == 1, 1, n == 4, 24, n == 7, 3408], True, If[n > 2, 24*Binomial[n, 5]*(n - 5)! /3^((n - 5)/3)*(n - 2)/3 + 2*Binomial[n, 2] * (n - 2)! /3^((n - 2)/3), 2]]; Array[a, 23] (* Jean-François Alcover, Dec 03 2017, after R. J. Mathar *) PROG (PARI) a(n) = { if (n == 1, return(1)); my(c=(n, k)->binomial(n, k), b=n->if(n>=0, n!/3^(n\3), 0), r=n%3); if (r == 0, b(n), r == 2, 24*c(n, 5)*b(n-5)*((n-2)\3) + 2*c(n, 2)*b(n-2), 720*c(n, 7)*((n-4)\3)*b(n-7) + 24^2*c(10, 5)*c(n, 10)*c((n-4)\3, 2)*b(n-10) + 6*c(n, 4)*b(n-4) + 18*c(n, 4)*b(n-4) + 48*c(n, 2)*c(n-2, 5)*b(n-7)*((n-4)\3)); }; vector(23, n, a(n)) \\ Gheorghe Coserea, Jun 02 2018 CROSSREFS Cf. A113583, A272640. Sequence in context: A261517 A131448 A156447 * A334123 A127261 A239531 Adjacent sequences: A272638 A272639 A272640 * A272642 A272643 A272644 KEYWORD nonn AUTHOR N. J. A. Sloane, May 06 2016 STATUS approved

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Last modified September 29 23:45 EDT 2023. Contains 365781 sequences. (Running on oeis4.)