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%I #48 Jun 14 2023 16:49:32
%S 1,3,31,1145,154881,77899563,147226330175,1053765855157617,
%T 28736455088578690945,3000127124463666294963283,
%U 1203831304687539089648950490463,1862632561783036151478238040096092649,11143500837236042423379349834982088594105985
%N Degree of the variety of pairs of commuting n X n matrices.
%C Also, ratio of vector elements of the ground state 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 These numbers arise in a similar way to A005130 and related sequences appear in the groundstate of the integrable Temperley-Lieb Hamiltonian.
%C It is also the weighted enumeration of lattice paths on an n X n square lattice going from the left side to the top side, with same initial and final orders of paths, and with a weight of 2 per vertex where a path turns 90 degrees. - _Paul Zinn-Justin_, Mar 05 2023
%H Paul Zinn-Justin, <a href="/A029729/b029729.txt">Table of n, a(n) for n = 1..16</a>
%H Jan de Gier, <a href="https://arxiv.org/abs/math/0211285">Loops, matchings and alternating-sign matrices</a>, arXiv:math/0211285 [math.CO], 2002-2003.
%H P. Di Francesco and P. Zinn-Justin, <a href="https://doi.org/10.1007/s00220-005-1476-5">Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties</a>, Comm. Math. Phys., 262(2):459-487, 2006; <a href="http://arxiv.org/abs/math-ph/0412031">arXiv preprint</a>, arXiv:math-ph/0412031, 2004-2005.
%H A. Garbali and P. Zinn-Justin, <a href="http://arxiv.org/abs/2110.07155">Shuffle algebras, lattice paths and the commuting scheme</a>, arXiv:2110.07155 [math.RT], 2021-2022. See also <a href="https://www.unimelb-macaulay2.cloud.edu.au/#tutorial-cirm-8">Macaulay2 code</a> to generate the sequence.
%H A. Knutson and P. Zinn-Justin, <a href="https://doi.org/10.1016/j.aim.2006.09.016">A scheme related to the Brauer loop model</a>, Adv. Math., 214(1):40-77, 2007, <a href="https://arxiv.org/abs/math/0503224">arXiv preprint</a>, arXiv:math/0503224 [math.AG], 2005-2006.
%H Macaulay 2 Manual, <a href="http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.8.2/share/Macaulay2/Macaulay2Doc/test/matrix.m2">Test of matrix routines</a>, Viewed May 03 2016.
%H M. J. Martins, B. Nienhuis, and R. Rietman, <a href="https://arxiv.org/abs/cond-mat/9709051">An Intersecting Loop Model as a Solvable Super Spin Chain</a>, arXiv:cond-mat/9709051 [cond-mat.stat-mech], 1997; Phys. Rev. Lett. Vol. 81 (1998) pp. 504-507.
%H Ada Stelzer and Alexander Yong, <a href="https://arxiv.org/abs/2306.00737">Combinatorial commutative algebra rules</a>, arXiv:2306.00737 [math.CO], 2023.
%F There is a formula in terms of divided differences operators (too complicated to reproduce here).
%e 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 (link above): 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 (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 Cf. A005130.
%K nonn,nice
%O 1,2
%A Nolan R. Wallach (nwallach(AT)euclid.ucsd.edu), Dec 11 1999
%E Entry revised based on comments from _Paul Zinn-Justin_, Mar 14 2005
%E Terms a(12) and beyond from _Paul Zinn-Justin_, Mar 05 2023
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