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A006932 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).

%I A006932 M2862 #71 Jul 02 2025 16:01:55
%S A006932 1,1,3,10,43,223,1364,9643,77545,699954,7013079,77261803,928420028,
%T A006932 12085410927,169413357149,2544367949634,40758600588283,
%U A006932 693684669653911,12499734669634036,237734433597317987,4759174459355303521
%N A006932 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).
%C A006932 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
%D A006932 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006932 Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 49.
%D A006932 K. Wayland, personal communication.
%H A006932 Vincenzo Librandi, <a href="/A006932/b006932.txt">Table of n, a(n) for n = 1..200</a>
%H A006932 Theresa Baren, Michael Cory, Mia Friedberg, Peter Gardner, James Hammer, Joshua Harrington, Daniel McGinnis, Riley Waechter, and Tony W. H. Wong, <a href="https://arxiv.org/abs/1810.03409">On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points</a>, arXiv:1810.03409 [math.CO], 2018.
%H A006932 Todd Feil, Gary Kennedy and David Callan, <a href="http://www.jstor.org/stable/2324797">Problem E3467</a>, Amer. Math. Monthly, 100 (1993), 800-801.
%H A006932 V. Strehl, <a href="/A003149/a003149.pdf">The average number of splitters in a random permutation</a> [Unpublished; included here with the author's permission.]
%F A006932 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
%F A006932 a(n) = Sum_{k=1..n} (n-k)!*A145878(k-1,0). See the link by Theresa Baren, et al. - _Daniel A. McGinnis_, Oct 15 2018
%F A006932 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
%F A006932 a(n) + A052186(n) = n! - _Pontus von Brömssen_, Jul 10 2021
%p A006932 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 Sellers_, Mar 13 2000
%t A006932 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 *)
%Y A006932 Cf. A003149, A052186, A145878.
%K A006932 nonn,easy,nice
%O A006932 1,3
%A A006932 _N. J. A. Sloane_
%E A006932 More terms from _James Sellers_, Mar 13 2000
%E A006932 Edited by _Emeric Deutsch_, Oct 29 2008