%I A005118 M2097
%S A005118 1,1,1,2,16,768,292864,1100742656,48608795688960,29258366996258488320,
%T A005118 273035280663535522487992320,44261486084874072183645699204710400,
%U A005118 138018895500079485095943559213817088756940800
%N A005118 Number of simple allowable sequences on 1..n containing the permutation 
               12...n.
%C A005118 For n >= 2 by the hook length formula a(n) is also the number of Young 
               tableaux of size 1+2+...+(n-1) = n(n-1)/2 that correspond to the 
               partition (1,2,...n-1), i.e. triangular Young tableaux. For example 
               when n=5 a(5)=768 and the shape of the tableau is xxxx / xxx / xx 
               / x. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 04 2001
%C A005118 Also, a(n) is the degree of the symplectic Grassmannian, the projective 
               variety of all maximal isotropic subspaces in a complex vector space 
               of dimension 2n-2 with a symplectic form. See Hiller's paper. - Burt 
               Totaro (b.totaro(AT)dpmms.cam.ac.uk), Oct 29 2002
%C A005118 Also, for n >= 2, a(n) is the number of maximal chains in the poset of 
               Dyck paths ordered by inclusion. - Jennifer Woodcock (Jennifer.Woodcock(AT)ugdsb.on.ca), 
               May 21 2008
%D A005118 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005118 J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational 
               Geometry, CRC Press, 1997, p. 102.
%D A005118 H. Hiller. Combinatorics and intersection of Schubert varieties. Comment. 
               Math. Helv. 57 (1982), 41-59.
%D A005118 G. Kreweras, Sur un probleme de scrutin a plus de deux candidats, Publications 
               de l'Institut de Statistique de l'Universit\'{e} de Paris, 26 (1981), 
               69-87.
%D A005118 R. P. Stanley, On the number of reduced decompositions of elements of 
               certain groups, European J. Combin., 5 (1984), 359-372.
%H A005118 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/comb.ps">
               A combinatorial miscellany</a>
%H A005118 R. P. Stanley, <a href="http://arXiv.org/abs/math.CO/0501256">Ordering 
               events in Minkowski space</a>
%F A005118 C(n, 2)!/(1^{n-1} * 3^{n-2} *...* (2n-3)^1 ).
%F A005118 a(n)=(n(n-1)/2)!/A057863(n-1) (n>=1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               May 21 2004
%F A005118 a(n) = A153452(A002110(n-1)). - Naohiro Nomoto, Jan 01 2009
%p A005118 A005118 := proc(n) local i; binomial(n,2)!/product( (2*i+1)^(n-i-1), 
               i=0..n-2 ); end;
%Y A005118 Cf. A003121, A018241.
%Y A005118 Cf. A057863.
%Y A005118 Sequence in context: A012464 A128294 A015188 this_sequence A108400 A013029 
               A012915
%Y A005118 Adjacent sequences: A005115 A005116 A005117 this_sequence A005119 A005120 
               A005121
%K A005118 nonn,easy,nice
%O A005118 0,4
%A A005118 N. J. A. Sloane (njas(AT)research.att.com).