A213090 Number of permutations of length n whose associated Schubert variety is defined by inclusions.
1, 1, 2, 6, 23, 101, 477, 2343, 11762, 59786, 306132, 1574536, 8120782, 41957030, 217021682, 1123371986, 5817788471, 30139492189, 156174965473, 809382185187, 4195096032623, 21745137658765, 112720985668763, 584336632836945, 3029232133574325, 15703985220888071
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..500
- H. Abe and S. Billey, Consequences of the Lakshmibai-Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry, 2014. See Th. 4.13.
- M. H. Albert and R. Brignall, Enumerating indices of Schubert varieties defined by inclusions, arXiv:1301.3188 [math.CO], 2013.
- V. Gasharov and V. Reiner, Cohomology of smooth Schubert varieties in partial flag manifolds, J. Lond. Math. Soc. 66 (2002), 550-562.
- A. Hultman, Inversion arrangements and Bruhat intervals, J. Combin. Theory Ser. A, 118(7) (2011), 1897-1906.
- A. Hultman, S. Linusson, J. Shareshian, and J. Sjöstrand, From Bruhat intervals to intersection lattices and a conjecture of Postnikov, J. Combin. Theory Ser. A, 116(3) (2009), 564-580.
- S. Oh, A. Postnikov and H. Yoo, Bruhat order, smooth Schubert varieties, and hyperplane arrangements, J. Combin. Theory Ser. A 115(7) (2008), 1156-1166.
- A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.
- Vic Reiner, Richard Stanley, and Joel Lewis, P0011 in the Database of Permutation Pattern Avoidance.
- J. Sjöstrand, Bruhat intervals as rooks on skew Ferrers boards, J. Combin. Theory Ser. A 114 (2007), 1182-1198.
Programs
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Mathematica
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 *)
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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
Formula
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
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
Comments