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

0,3

Comments

Permutations avoiding the four permutation patterns 4231, 35142, 42513, 351624.

See references for several other characterizations.

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, 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.

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A133656 A078487 A193038 * A218225 A279572 A263576

Adjacent sequences: A213087 A213088 A213089 * A213091 A213092 A213093

Keyword

nonn

Author

Joel B. Lewis, Jun 05 2012

Status

approved