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A213090 Number of permutations of length n whose associated Schubert variety is defined by inclusions.

%I A213090 #42 Mar 12 2025 15:39:55
%S A213090 1,1,2,6,23,101,477,2343,11762,59786,306132,1574536,8120782,41957030,
%T A213090 217021682,1123371986,5817788471,30139492189,156174965473,
%U A213090 809382185187,4195096032623,21745137658765,112720985668763,584336632836945,3029232133574325,15703985220888071
%N A213090 Number of permutations of length n whose associated Schubert variety is defined by inclusions.
%C A213090 Permutations avoiding the four permutation patterns 4231, 35142, 42513, 351624.
%C A213090 See references for several other characterizations.
%H A213090 Alois P. Heinz, <a href="/A213090/b213090.txt">Table of n, a(n) for n = 0..500</a>
%H A213090 H. Abe and S. Billey, <a href="http://www.math.washington.edu/~billey/papers/abe.billey.pdf">Consequences of the Lakshmibai-Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry</a>, 2014. See Th. 4.13.
%H A213090 M. H. Albert and R. Brignall, <a href="http://arxiv.org/abs/1301.3188">Enumerating indices of Schubert varieties defined by inclusions</a>, arXiv:1301.3188 [math.CO], 2013.
%H A213090 V. Gasharov and V. Reiner, <a href="http://dx.doi.org/10.1112/S0024610702003605">Cohomology of smooth Schubert varieties in partial flag manifolds</a>, J. Lond. Math. Soc. 66 (2002), 550-562.
%H A213090 A. Hultman, <a href="https://doi.org/10.1016/j.jcta.2011.04.005">Inversion arrangements and Bruhat intervals</a>, J. Combin. Theory Ser. A, 118(7) (2011), 1897-1906.
%H A213090 A. Hultman, S. Linusson, J. Shareshian, and J. Sjöstrand, <a href="http://dx.doi.org/10.1016/j.jcta.2008.09.001">From Bruhat intervals to intersection lattices and a conjecture of Postnikov</a>, J. Combin. Theory Ser. A, 116(3) (2009), 564-580.
%H A213090 S. Oh, A. Postnikov and H. Yoo, <a href="http://dx.doi.org/10.1016/j.jcta.2008.01.003">Bruhat order, smooth Schubert varieties, and hyperplane arrangements</a>, J. Combin. Theory Ser. A 115(7) (2008), 1156-1166.
%H A213090 A. Postnikov, <a href="http://arxiv.org/abs/math/0609764">Total positivity, Grassmannians, and networks</a>, arXiv:math/0609764 [math.CO], 2006.
%H A213090 Vic Reiner, Richard Stanley, and Joel Lewis, <a href="http://math.depaul.edu/bridget/cgi-bin/dppa.cgi?choice=3&amp;search=P0011">P0011</a> in the Database of Permutation Pattern Avoidance.
%H A213090 J. Sjöstrand, <a href="http://dx.doi.org/10.1016/j.jcta.2007.01.001">Bruhat intervals as rooks on skew Ferrers boards</a>, J. Combin. Theory Ser. A 114 (2007), 1182-1198.
%F A213090 G.f.: 1 + (1-3*x-2*x^2-(1-x-2*x^2)*sqrt(1-4*x)) / (1-3*x-(1-x+2*x^2) * sqrt(1-4*x)). - _Michael Albert_, Jan 15 2013
%F A213090 D-finite with recurrence n*a(n) +(-15*n+16)*a(n-1) +(77*n-158)*a(n-2) +(-149*n+408)*a(n-3) +2*(39*n-55)*a(n-4) +4*(-8*n+7)*a(n-5) +16*(-2*n+11)*a(n-6)=0. - _R. J. Mathar_, May 30 2014
%t A213090 1 + ((1 - 5x - 2x^2 + 8x^3) - Sqrt[1-4x] (1 - 5x - 2x^2))/(2(1 - 6x + 5x^2 - 4x^3)) + O[x]^26 // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 28 2018 *)
%o A213090 (PARI) (1-3*x-2*x^2-(1-x-2*x^2)*sqrt(1-4*x))/(1-3*x-(1-x+2*x^2)*sqrt(1-4*x)) \\ _Charles R Greathouse IV_, Oct 20 2015
%K A213090 nonn
%O A213090 0,3
%A A213090 _Joel B. Lewis_, Jun 05 2012