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A006932
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Number of permutations of [n] with at least one strong fixed point (a permutation p of {1,2,...,n} is said to have j as a strong fixed point if p(k) < j for k < j and p(k) > j for k > j).
(Formerly M2862)
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6
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1, 1, 3, 10, 43, 223, 1364, 9643, 77545, 699954, 7013079, 77261803, 928420028, 12085410927, 169413357149, 2544367949634, 40758600588283, 693684669653911, 12499734669634036, 237734433597317987, 4759174459355303521
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OFFSET
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1,3
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COMMENTS
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a(n) is also the number of permutation graphs with domination number one. See Definition 2.1, Lemma 2.3, and page 16 in the paper provided in the link by Theresa Baren, et al. - Daniel A. McGinnis, Oct 16 2018
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 49.
K. Wayland, personal communication.
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LINKS
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Theresa Baren, Michael Cory, Mia Friedberg, Peter Gardner, James Hammer, Joshua Harrington, Daniel McGinnis, Riley Waechter, and Tony W. H. Wong, On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points, arXiv:1810.03409 [math.CO], 2018.
Todd Feil, Gary Kennedy and David Callan, Problem E3467, Amer. Math. Monthly, 100 (1993), 800-801.
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FORMULA
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a(n) ~ 2 * (n-1)! * (1 - 1/(2*n) + 1/(2*n^2) + 9/(2*n^3) + 59/(2*n^4) + 237/n^5 + 2280/n^6 + 25182/n^7 + 625385/(2*n^8) + 4311329/n^9 + 65375943/n^10). - Vaclav Kotesovec, Mar 17 2015
a(n) = A003149(n-1) - Sum_{k=0..n-1} (n-k-1)!*a(k). (This follows immediately from the preceding formula since A145878(k,0) = k! - a(k).) - Pontus von Brömssen, Jul 10 2021
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MAPLE
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t1 := sum(n!*x^n, n=0..100): F := series(t1/(1+x*t1), x, 100): for i from 1 to 40 do printf(`%d, `, i!-coeff(F, x, i)) od: # James A. Sellers, Mar 13 2000
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MATHEMATICA
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m = 22; s = Sum[n!*x^n, {n, 0, m}]; Range[0, m-1]! - CoefficientList[ Series[ s/(1+x*s), {x, 0, m}], x][[1;; m]] // Rest (* Jean-François Alcover, Apr 28 2011, after Maple code *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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