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

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, 1907, 572, 22, 1, 84, 1428, 7656, 15311, 12004, 3514, 312, 10, 1, 120, 3024, 25332, 85543, 127804, 88034, 28296, 4342, 368, 16

Offset: 1

Author

Emeric Deutsch, May 28 2008

Keywords

• nonn
• tabf

Comments

Definition of white corners (as used in the Eriksson/Linusson reference):

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.

Equivalent definitions can use different borders and orientations.

Sum of entries in row n is n! (A000142).

Sum(k*T(n,k),k=0..max(k))=A140712(n).

This triangle is irregular, its length grows slightly faster than n.

Examples

			Triangle starts:
1;
1,1;
1,4,1;
1,10,12,1;
1,20,62,36,1;
1,35,217,339,126,2;
1,56,602,1880,1907,572,22;
1,84,1428,7656,15311,12004,3514,312,10;
1,120,3024,25332,85543,127804,88034,28296,4342,368,16;
		

References

• K. Eriksson and S. Linusson. Combinatorics of Fulton's essential set. Duke Mathematical Journal 85(1):61-76, 1996.
• W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), 381-420.
• I. G. Macdonald, Notes on Schubert polynomials, Département de mathématiques et d’informatique, Université du Québec, Montréal, 1991.

Links

• K. Eriksson and S. Linusson, The size of Fulton's essential set, Electronic J. Combinatorics, Vol. 2, #R6, 1995.
• K. Eriksson and S. Linusson, Combinatorics of Fulton's essential set, ResearchGate, 1998.

Crossrefs

Cf. A000142, A140712.

Cf. A213166 (permutations by white global corners).

Extensions

Edited and extended by Olivier Gérard, Oct 30 2012