A111111 Number of simple permutations of degree n.

1, 2, 0, 2, 6, 46, 338, 2926, 28146, 298526, 3454434, 43286526, 583835650, 8433987582, 129941213186, 2127349165822, 36889047574274, 675548628690430, 13030733384956418, 264111424634864638, 5612437196153963522, 124789500579376435198, 2897684052921851965442

Offset: 1

Author

N. J. A. Sloane, Oct 14 2005

Keywords

• nonn
• nice

Comments

A permutation is simple if the only intervals that are mapped onto intervals are the singletons and [1..n].

For example, the permutation

1234567

2647513

is not simple since it maps [2..5] onto [4..7].

In other words, a permutation [1 ... n] -> [p_1 p_2 ... p_n] is simple if there is no string of consecutive numbers [i_1 ... i_k] which is mapped onto a string of consecutive numbers [p_i_1 ... p_i_k] except for the strings of length k = 1 or n.

Examples

			G.f. = x + 2*x^2 + 2*x^4 + 6*x^5 + 46*x^6 + 338*x^7 + 2926*x^8 + ...
The simple permutations of lowest degree are 1, 12, 21, 2413, 3142.
		

References

• Corteel, Sylvie; Louchard, Guy; and Pemantle, Robin, Common intervals of permutations. in Mathematics and Computer Science. III, 3--14, Trends Math., Birkhuser, Basel, 2004.
• S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7
• Bridget Eileen Tenner, Interval posets of permutations, arXiv:2007.06142, Aug 2021.

Links

• T. D. Noe, Table of n, a(n) for n = 1..100
• M. H. Albert and M. D. Atkinson, Simple permutations and pattern restricted permutations, Discr. Math., 300 (2005), 1-15.
• M. H. Albert, M. D. Atkinson and M. Klazar, The enumeration of simple permutations, Journal of Integer Sequences 6 (2003), Article 03.4.4, 18 pages.
• Joerg Arndt, All simple permutations for n <= 6
• Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.
• M. Bouvel, M. Mishna, and C. Nicaud, Some simple varieties of trees arising in permutation analysis, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 855-866.
• Robert Brignall, Sophie Huczynska, and Vincent Vatter, Simple permutations and algebraic generating functions, arXiv:math/0608391 [math.CO], (2006).
• R. Brignall, S. Huczynska, and V. Vatter, Decomposing simple permutations with enumerative consequences, Combinatorica, 28 (2008) 384-400.
• Robert Brignall, A Survey of Simple Permutations, arXiv:0801.0963 [math.CO], (18-April-2008)
• Sylvie Corteel, Guy Louchard, and Robin Pemantle, Common intervals in permutations, Discrete Math. Theor. Comput. Sci. 8 (2006), no. 1, 189-216.
• Scott Garrabrant and Igor Pak, Pattern Avoidance is Not P-Recursive, preprint, 2015.
• V. Jelínek and P. Valtr, Splittings and Ramsey Properties of Permutation Classes, arXiv preprint arXiv:1307.0027 [math.CO], 2013.
• Djamila Oudrar, Sur l'énumération de structures discrètes, une approche par la théorie des relations, Thesis (in French), arXiv:1604.05839 [math.CO], 2016.
• Djamila Oudrar and Maurice Pouzet, Profile and hereditary classes of ordered relational structures, arXiv preprint arXiv:1409.1108 [math.CO], 2014.
• Djamila Oudrar, Maurice Pouzet, and Imed Zaguia, Minimal prime ages, words and permutation graphs Extended abstract, arXiv:2205.08992 [math.CO], 2022.

Crossrefs

Cf. A059372, A280780.

Programs

Mathematica

nmax = 20; t[n_, k_] := t[n, k] = Sum[(m + 1)!*t[n - m - 1, k - 1], {m, 0, n - k}]; t[n_, 1] = n!; t[n_, n_] = 1; tnk = Table[t[n, k], {n, 1, nmax}, {k, 1, nmax}]; A111111 = -Inverse[tnk][[All, 1]] + 2*(-1)^Range[0, nmax - 1]; A111111[[2]] = 2;
A111111 (* Jean-François Alcover, Jul 13 2016 *)

PARI

simple(v)=for(i=1,#v-1, for(j=i+1,#v, my(u=vecsort(v[i..j]));if(u[#u]-u[1]==#u-1 && #u<#v, return(0)))); 1
a(n)=sum(i=0,n!-1, simple(numtoperm(n,i))) \\ Charles R Greathouse IV, Nov 05 2013
seq(N) = Vec(2 + 2*x^2 - 2/(1+x) - serreverse(x*Ser(vector(N, n, n!))));  \\ Gheorghe Coserea, Jan 22 2017

Formula

a(n) = -A059372(n)+2(-1)^(n+1).

a(n) ~ n!*(1-4/n)/e^2. - Jon E. Schoenfield, Aug 05 2006

a(n) ~ n!*exp(-2)*(1 - 4/n + 2/(n*(n-1)) - (40/3)/(n*(n-1)*(n-2)) - ...). Coefficients are given by A280780(n)/A280781(n).- Gheorghe Coserea, Jan 23 2017

Extensions

Incorrect statement removed by Jay Pantone, Jul 16 2014

Word "fixed" removed by Franklin T. Adams-Watters, Jul 22 2014