%I A029729
%S A029729 1,3,31,1145,154881,77899563,147226330175,1053765855157617,
%T A029729 28736455088578690945,3000127124463666294963283,
%U A029729 1203831304687539089648950490463
%N A029729 Degree of the variety of pairs of commuting n X n matrices.
%C A029729 Also, ratio of vector elements of the groundstate in the loop representation 
               of the braid-monoid Hamiltonian H = Sum_i (3 - 2 e_i - b_i) with 
               size 2n and periodic boundary conditions. Specifically the smallest 
               element that corresponds to a non-crossing chord diagram, divided 
               by the overall smallest element. We reduce the standard braid-monoid 
               algebra to the Brauer algebra B_{2n}(1). - B. Nienhuis & J. de Gier 
               (B.Nienhuis(AT)UvA.NL), May 13 2004. For a proof that this is the 
               same sequence, see the articles by P. Di Francesco and P. Zinn-Justin 
               and A. Knutson and P. Zinn-Justin.
%C A029729 These numbers arise in a similar way to A005130 and related sequences 
               appear in the groundstate of the integrable Temperley-Lieb Hamiltonian.
%D A029729 P. Di Francesco, P. Zinn-Justin, Inhomogeneous model of crossing loops 
               and multidegrees of some algebraic varieties, preprint math-ph/0412031
%D A029729 A. Knutson, P. Zinn-Justin, A scheme related to the Brauer loop model, 
               preprint math.AG/0503224
%D A029729 M. J. Martins, B. Nienhuis, R. Rietman, An Intersecting Loop Model as 
               a Solvable Super Spin Chain, Phys.Rev.Lett. Vol. 81 (1998) pp. 504-507
%D A029729 A. V. Razumov, Yu. G. Stroganov, Combinatorial nature of ground state 
               vector of O(1) loop model, Theor.Math.Phys. 138 (2004) pp. 333-337; 
               Teor.Mat.Fiz. 138 (2004) pp. 395-400
%H A029729 Jan de Gier, <a href="http://lanl.arXiv.org/abs/math.CO/0211285">Loops, 
               matchings and alternating-sign matrices</a>
%H A029729 P. Di Francesco, P. Zinn-Justin, <a href="http://arXiv.org/abs/math-ph/
               0412031">Inhomogeneous model of crossing loops and multidegrees of 
               some algebraic varieties</a>
%H A029729 A. Knutson, P. Zinn-Justin, <a href="http://arXiv.org/abs/math.AG/0503224">
               A scheme related to the Brauer loop model</a>
%F A029729 There is a formula in terms of divided differences operators (too complicated 
               to reproduce here).
%e A029729 n=1: Degree of C X C which is 1. n=2: The degree can be calculated by 
               hand to be 3. n=3: See Macaulay manual: one of steps in proof that 
               variety for 3 X 3 is Cohen-Macaulay is to compute the degree which 
               is 31. (n=4) Matt Clegg (CS at UCSD) and Nolan Wallach using 10 Sun 
               Workstations and a distributed Grobner Basis package (in 1993).
%e A029729 (2(e1 + e2 + e3 + e4) + b1 + b2 + b3 + b4)(G + G e2 + b2)(e1 e3 b2) = 
               12 (G + G e2 + b2)(e1 e3 b2) with G = 3, therefore a(2) = 3
%Y A029729 Cf. A005130.
%Y A029729 Sequence in context: A144416 A022514 A094579 this_sequence A136584 A141153 
               A144906
%Y A029729 Adjacent sequences: A029726 A029727 A029728 this_sequence A029730 A029731 
               A029732
%K A029729 nonn,nice
%O A029729 1,2
%A A029729 Nolan R. Wallach (nwallach(AT)euclid.ucsd.edu)
%E A029729 Entry revised based on comments from P. Zinn-Justin (pzinn(AT)lptms.u-psud.fr), 
               Mar 14 2005