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A140711 Irregular triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k white corners.

%I A140711 #12 Nov 28 2018 12:16:51
%S A140711 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 A140711 1907,572,22,1,84,1428,7656,15311,12004,3514,312,10,1,120,3024,25332,
%U A140711 85543,127804,88034,28296,4342,368,16
%N A140711 Irregular triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k white corners.
%C A140711 Definition of white corners (as used in the Eriksson/Linusson reference):
%C A140711 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 A140711 Equivalent definitions can use different borders and orientations.
%C A140711 Sum of entries in row n is n! (A000142).
%C A140711 Sum(k*T(n,k),k=0..max(k))=A140712(n).
%C A140711 This triangle is irregular, its length grows slightly faster than n.
%D A140711 K. Eriksson and S. Linusson. Combinatorics of Fulton's essential set. Duke Mathematical Journal 85(1):61-76, 1996.
%D A140711 W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), 381-420.
%D A140711 I. G. Macdonald, Notes on Schubert polynomials, Département de mathématiques et d’informatique, Université du Québec, Montréal, 1991.
%H A140711 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 A140711 K. Eriksson and S. Linusson, <a href="https://www.researchgate.net/publication/2550835_Combinatorics_of_Fulton&#39;s_essential_set">Combinatorics of Fulton's essential set</a>, ResearchGate, 1998.
%e A140711 Triangle starts:
%e A140711 1;
%e A140711 1,1;
%e A140711 1,4,1;
%e A140711 1,10,12,1;
%e A140711 1,20,62,36,1;
%e A140711 1,35,217,339,126,2;
%e A140711 1,56,602,1880,1907,572,22;
%e A140711 1,84,1428,7656,15311,12004,3514,312,10;
%e A140711 1,120,3024,25332,85543,127804,88034,28296,4342,368,16;
%Y A140711 Cf. A000142, A140712.
%Y A140711 Cf. A213166 (permutations by white global corners).
%K A140711 nonn,tabf
%O A140711 1,5
%A A140711 _Emeric Deutsch_, May 28 2008
%E A140711 Edited and extended by _Olivier Gérard_, Oct 30 2012