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%I #12 Nov 28 2018 12:16:51
%S 1,1,1,1,4,1,1,10,12,1,1,20,62,36,1,1,35,217,339,126,2,1,56,602,1880,
%T 1907,572,22,1,84,1428,7656,15311,12004,3514,312,10,1,120,3024,25332,
%U 85543,127804,88034,28296,4342,368,16
%N Irregular triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k white corners.
%C Definition of white corners (as used in the Eriksson/Linusson reference):
%C In the representation of a permutation p as a n*n square array with n black cells at positions (i,p(i)), color in gray all cells in the row segment from each black cell to the right (east) border and in the column segment from each black cell to the bottom (south) border. Among the remaining white cells, the white corners are those without east or south white neighbors.
%C Equivalent definitions can use different borders and orientations.
%C Sum of entries in row n is n! (A000142).
%C Sum(k*T(n,k),k=0..max(k))=A140712(n).
%C This triangle is irregular, its length grows slightly faster than n.
%D K. Eriksson and S. Linusson. Combinatorics of Fulton's essential set. Duke Mathematical Journal 85(1):61-76, 1996.
%D W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), 381-420.
%D I. G. Macdonald, Notes on Schubert polynomials, Département de mathématiques et d’informatique, Université du Québec, Montréal, 1991.
%H K. Eriksson and S. Linusson, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v2i1r6">The size of Fulton's essential set</a>, Electronic J. Combinatorics, Vol. 2, #R6, 1995.
%H K. Eriksson and S. Linusson, <a href="https://www.researchgate.net/publication/2550835_Combinatorics_of_Fulton's_essential_set">Combinatorics of Fulton's essential set</a>, ResearchGate, 1998.
%e Triangle starts:
%e 1;
%e 1,1;
%e 1,4,1;
%e 1,10,12,1;
%e 1,20,62,36,1;
%e 1,35,217,339,126,2;
%e 1,56,602,1880,1907,572,22;
%e 1,84,1428,7656,15311,12004,3514,312,10;
%e 1,120,3024,25332,85543,127804,88034,28296,4342,368,16;
%Y Cf. A000142, A140712.
%Y Cf. A213166 (permutations by white global corners).
%K nonn,tabf
%O 1,5
%A _Emeric Deutsch_, May 28 2008
%E Edited and extended by _Olivier Gérard_, Oct 30 2012
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