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

1,5

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.

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

Table of n, a(n) for n=1..48.

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.

Example

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;

Crossrefs

Cf. A000142, A140712.

Cf. A213166 (permutations by white global corners).

Sequence in context: A202924 A142595 A174669 * A164366 A121692 A264614

Adjacent sequences: A140708 A140709 A140710 * A140712 A140713 A140714

Keyword

nonn,tabf

Author

Emeric Deutsch, May 28 2008

Extensions

Edited and extended by Olivier Gérard, Oct 30 2012

Status

approved