OFFSET
1,2
COMMENTS
Also the evaluation of the sum over all permutations w in S_n of the corresponding Schubert polynomial with all variables set equal to 1.
REFERENCES
I. G. Macdonald, Notes on Schubert Polynomials, Publications du LACIM 6, Université du Québec à Montréal, 1991.
LINKS
N. Bergeron and S. Billey, RC-Graphs and Schubert polynomials, Experimental Mathematics, Vol.2, No. 4, 1993.
S. Billey, W. Jockusch and R. P. Stanley, Some combinatorial properties of Schubert polynomials, Journal of Algebraic Combinatorics 2(4):345-374, 1993.
S. Fomin and A. N. Kirillov, Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Mathematics, Volume 153, Issues 1-3, 1 June 1996, Pages 123-143; Proceedings of the Conference on Power Series and Algebraic Combinatorics, Firenze, 1993.
S. Fomin and R. Stanley, Schubert Polynomials and the NilCoxeter algebra, Adv. Math. 103, 1994.
A. Lascoux and M.-P. Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris 294, 447-450, 1982.
A. Morales, I. Pak and G. Panova, Asymptotics of principal evaluations of Schubert polynomials for layered permutations, arxiv:1805.04341 [math.CO], 2018; Proceedings of the American Mathematical Society, Vol 147 1377-1389, 2019.
R. P. Stanley, Some Schubert Shenanigans, arXiv:1704.00851 [math.CO], 2017.
A. Weigandt, Bumpless pipe dreams and alternating sign matrices, arXiv:2003.07342 [math.CO], 2020.
FORMULA
From Alejandro H. Morales, Aug 29 2020: (Start)
log_2(a(n))/n^2 -> c (conjectured by Stanley in arxiv:1704.00851).
c >= 0.293236... defined as max(f(x) + c*x^2, 0<=x<1) = c, where f(x) = x^2*log(x) - (1/2)*(1-x)^2*log(1-x) - (1/2)*(1+x)^2*log(1+x) + 2*x*log(2) (proved in arxiv:1805.04341).
c <= 0.37 (observation as a corollary of a result in arXiv:2003.07342).
(End)
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Sara Billey, Feb 01 2020
STATUS
approved