1,2

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.

These numbers arise in a similar way to A005130 and related sequences appear in the groundstate of the integrable Temperley-Lieb Hamiltonian.

P. Di Francesco, P. Zinn-Justin, Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties, preprint math-ph/0412031

A. Knutson, P. Zinn-Justin, A scheme related to the Brauer loop model, preprint math.AG/0503224

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

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

Jan de Gier, Loops, matchings and alternating-sign matrices

P. Di Francesco, P. Zinn-Justin, Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties

A. Knutson, P. Zinn-Justin, A scheme related to the Brauer loop model

There is a formula in terms of divided differences operators (too complicated to reproduce here).

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).

(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

Cf. A005130.

Sequence in context: A136370 A022514 A094579 this_sequence A136584 A081789 A066976

Adjacent sequences: A029726 A029727 A029728 this_sequence A029730 A029731 A029732

nonn,nice

Nolan R. Wallach (nwallach(AT)euclid.ucsd.edu)

Entry revised based on comments from P. Zinn-Justin (pzinn(AT)lptms.u-psud.fr), Mar 14 2005