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A213090 Number of permutations of length n whose associated Schubert variety is defined by inclusions. 1
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 (list; graph; refs; listen; history; text; internal format)
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

Conjecture: 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

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Last modified September 20 23:04 EDT 2021. Contains 347596 sequences. (Running on oeis4.)