Search: id:A003319
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%I A003319 M2948
%S A003319 1,1,3,13,71,461,3447,29093,273343,2829325,31998903,392743957,
%T A003319 5201061455,73943424413,1123596277863,18176728317413,311951144828863,
%U A003319 5661698774848621,108355864447215063,2181096921557783605
%N A003319 Number of connected permutations of [1..n] (those not fixing [1..j] for 
               0<j<n). Also called indecomposable permutations.
%C A003319 Also the number of permutations with no global descents, as introduced 
               by Aguiar and Sottile [Corollaries 6.3, 6.4 and Remark 6.5]
%C A003319 Also the dimensions of the homogeneous components of the space of primitive 
               elements of the Malvenuto-Reutenauer Hopf algebra of permutations. 
               This result, due to Poirier and Reutenauer [Theoreme 2.1] is stated 
               in this form in the work of Aguiar and Sottile [Corollary 6.3] and 
               also in the work of Duchamp, Hivert and Thibon [Section 3.3]
%C A003319 Related to number of subgroups of index n-1 in free group of rank 2 (i.e. 
               maximal number of subgroups of index n-1 in any 2-generator group). 
               See Problem 5.13(b) in Stanley's Enumerative Combinatorics, Vol. 
               2.
%C A003319 Left border of triangle A144107 = A003319, with row sums = n!. [From 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 11 2008]
%C A003319 Hankel transform is A059332. Hankel transform of aerated sequence is 
               A137704(n+1). [From Paul Barry (pbarry(AT)wit.ie), Oct 07 2008]
%C A003319 for every n, a(n+1) is also the moment of order n for the probability 
               density function rho(x)=exp(x)/(Ei(1,-x)*(Ei(1,-x)+2*I*Pi)) on the 
               interval 0..infinity, with Ei the exponentiel-integral function. 
               [From Groux Roland (roland.groux(AT)orange.fr), Jan 16 2009]
%D A003319 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003319 Marcelo Aguiar (Texas A&M University) and Frank Sottile (University of 
               Massachusetts at Amherst). math.CO/0203282 Structure of the Malvenuto-Reutenauer 
               Hopf algebra of permutations.
%D A003319 F. R. K. Chung and R. L. Graham, Primitive juggling sequences, preprint, 
               2006.
%D A003319 L. Comtet, Sur les coefficients de l'inverse de la series formelle Sum 
               n! t^n, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.
%D A003319 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 84 (#25), 262 (#14) 
               and 295 (#16).
%D A003319 J. D. Dixon, Asymptotics of generating the symmetric and alternating 
               groups, Electron. J. Combin., Item R56 of Volume 12(1), 2005.
%D A003319 G. Duchamp (University of Rouen), F. Hivert and J.-Y. Thibon (University 
               of Marne-la-Vallee). math.CO/0105065 Noncommutative symmetric functions 
               VI: Free quasi-symmetric functions and related algebras.
%D A003319 I. M. Gessel and R. P. Stanley, Algebraic Enumeration, chapter 21 in 
               Handbook of Combinatorics, Vol. 2, edited by R. L. Graham et al., 
               The MIT Press, Mass, 1995.
%D A003319 A. King, Generating indecomposable permutations, Discrete Math., 306 
               (2006), 508-518.
%D A003319 P. Ossona de Mendez and P. Rosenstiehl, Transitivity and connectivity 
               of permutations, Combinatorics, 24 (No. 3, 2004), 487-501.
%D A003319 L. Panaitopol, A formula for $\pi(x)$ applied to a result of Koninck-Ivi\'c, 
               Nieuw Arch. Wisk. 5/1 55-56 (2000)
%D A003319 S. Poirier and C. Reutenauer, Algebres Hopf de tableaux, Ann. Sci. Math. 
               Quebec 19 (95), no. 1, 79-90.
%D A003319 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 5.13(b).
%D A003319 R. P. Stanley, The Descent Set and Connectivity Set of a Permutation, 
               Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.8.
%D A003319 J. D. Dixon, The probability of generating the symmetric group, Math. 
               Z. 110 (1969) 199-205.
%H A003319 T. D. Noe, <a href="b003319.txt">Table of n, a(n) for n = 1..101</a>
%H A003319 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A003319 David Callan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting 
               Stabilized-Interval-Free Permutations</a>, Journal of Integer Sequences, 
               Vol. 7 (2004), Article 04.1.8.
%H A003319 I. M. Gessel and R. P. Stanley <a href="http://people.brandeis.edu/~gessel/
               homepage/papers/enum.pdf">Algebraic Enumeration</a> (See pages 7-8 
               for generating function.)
%H A003319 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               90
%F A003319 G.f.: 1-1/Sum (k! x^k ). Also a(n) = n! - Sum_{k=1..n-1} k!*a(n-k), n 
               >= 1.
%F A003319 a(n) = (-1)^{n-1} * det {| 1! 2! ... n! | 1 1! ... (n-1)! | 0 1 1! ... 
               (n-2)! | ... | 0 ... 0 1 1! |}
%F A003319 INVERTi transform of factorial numbers, A000142 starting from n=1. - 
               Antti Karttunen (Antti.Karttunen(AT)iki.fi), May 30 2003
%F A003319 Gives the row sums of the triangle [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] 
               DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the 
               operator defined in A084938; this triangle, read by rows is the sequence 
               : 1; 0, 1; 0, 1, 2; 0, 1, 6, 6; 0, 1, 12, 34, 24; 0, 1, 20, 110, 
               210, 120; 0, 1, 30, 270, 974, 1452, 720; ... - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Dec 30 2003
%F A003319 a(n+1)=Sum_{k,0<=k<=n}A089949(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 16 2006
%F A003319 L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=0} n!*x^n ) . - Paul D. 
               Hanna (pauldhanna(AT)juno.com), Sep 19 2007
%F A003319 G.f.: 1/(1-x/(1-2x/(1-2x/(1-3x/(1-3x/(1-4x/(1-4x/(1-.....))))))) (continued 
               fraction); [From Paul Barry (pbarry(AT)wit.ie), Oct 07 2008]
%F A003319 For n > 0 let R be the n-th row of A090238. Then a(n) = Sum{i=0..n}(-1)^(i)*R[i]. 
               [From Peter Luschny (peter(AT)luschny.de), Mar 13 2009]
%p A003319 INVERTi([seq(n!,n=1..20)]);
%Y A003319 Leading diagonal of A059438.
%Y A003319 Cf. A051296, A084938, A074664, A113869.
%Y A003319 A144107 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 11 2008]
%Y A003319 Sequence in context: A162326 A122455 A126390 this_sequence A158882 A000261 
               A111140
%Y A003319 Adjacent sequences: A003316 A003317 A003318 this_sequence A003320 A003321 
               A003322
%K A003319 nonn,easy,nice
%O A003319 1,3
%A A003319 N. J. A. Sloane (njas(AT)research.att.com).
%E A003319 More terms from Michael Somos, Jan 26 2000
%E A003319 Additional comments from Marcelo Aguiar (maguiar(AT)math.tamu.edu), Mar 
               28 2002

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