A140711 -id:A140711 - cp's OEIS frontend

A140712 Number of white corners in all permutations of {1,2,...,n} (for definition see the Eriksson-Linusson references).

0, 1, 6, 37, 256, 2000, 17520, 170520, 1827840, 21409920, 272160000, 3732220800, 54925516800, 863480217600, 14442536908800, 256086230400000, 4798293147648000, 94731418349568000, 1965528727658496000

Offset: 1

Emeric Deutsch, May 28 2008

K. Eriksson and S. Linusson. Combinatorics of Fulton's essential set. Duke Mathematical Journal 85(1):61-76, 1996.

G. C. Greubel, Table of n, a(n) for n = 1..445
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.

Cf. A000142, A140711.

List([1..30], n -> Factorial(n-1)*(n-1)*(n^2 +13*n+6)/36); # G. C. Greubel, Nov 28 2018

[Factorial(n-1)*(n-1)*(n^2+13*n+6)/36: n in [1..20]]; // G. C. Greubel, Nov 28 2018

seq((1/36)*(n-1)*factorial(n-1)*(n^2+13*n+6),n=1..20);

Table[(n-1)!*(n-1)*(n^2 +13*n+6)/36, {n,1,20}] (* G. C. Greubel, Nov 28 2018 *)

a(n)=(n^2+13*n+6)*(n-1)*(n-1)!/36 \\ Charles R Greathouse IV, Oct 31 2011

[factorial(n-1)*(n-1)*(n^2 +13*n+6)/36 for n in (1..20)] # G. C. Greubel, Nov 28 2018

a(n) = Sum_{k=0..n-1} k*A140711(n,k).
a(n) = (n-1)!*(binomial(n-1,3)+6*binomial(n,2))/6 = (n-1)!*(n-1)*(n^2+13*n+6)/36.
E.g.f.: (x*(6 + 3*x - 7*x^2) + (6 - 18*x + 18*x^2 - 6*x^3)*log(1-x))/(36* (1-x)^3). - G. C. Greubel, Nov 28 2018
D-finite with recurrence 2*(-n+5)*a(n) +(2*n^2-31*n+51)*a(n-1) +(25*n-14)*(n-2)*a(n-2)=0. - R. J. Mathar, Jul 26 2022

A140713 Triangle read by rows: T(n,k) is the number of white corners of rank k in all the permutations of {1,2,...,n} (n>=2, 0<=k<=n-2; for definitions see the Eriksson-Linusson references).

Original entry on oeis.org

1, 5, 1, 26, 9, 2, 154, 70, 26, 6, 1044, 562, 268, 102, 24, 8028, 4860, 2700, 1308, 504, 120, 69264, 45756, 28224, 15828, 7728, 3000, 720

Offset: 2

Emeric Deutsch, May 29 2008

Sum of row n is A140712(n).
T(n,0) = A001705(n-1).
T(n,n-2) = (n-2)!.

			Triangle starts:
1;
5,1;
26,9,2;
154,70,26,6;
1044,562,268,102,24;

K. Eriksson and S. Linusson. Combinatorics of Fulton's essential set. Duke Mathematical Journal 85(1):61-76, 1996.

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.

Cf. A001705, A140711, A140712.

A213166 Triangle, read by rows, of permutations of length n with k white global corners.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 10, 1, 1, 35, 63, 20, 1, 1, 111, 348, 224, 35, 1, 1, 405, 1920, 2027, 630, 56, 1, 1, 1733, 11247, 17142, 8600, 1512, 84, 1, 1, 8666, 71861, 145375, 104175, 29447, 3234, 120, 1

Offset: 1

Olivier Gérard, Oct 30 2012

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 (immediate) neighbors.
In this modified sequence (not present in the Eriksson/Linusson reference), white global corners are without east or south white neighbors in the east row segment and the south column segment joining the border. This ensures that there can be at most only one white global corner for a given row or a given column. The table is triangular.
Equivalent definitions can use different borders and orientations.
Sum of entries in row n is n! (A000142).

			Table begins:
1;
1, 1;
1, 4,    1;
1, 12,   10,    1;
1, 35,   63,    20,     1;
1, 111,  348,   224,    35,     1;
1, 405,  1920,  2027,   630,    56,    1;
1, 1733, 11247, 17142,  8600,   1512,  84,   1;
1, 8666, 71861, 145375, 104175, 29447, 3234, 120, 1;
...

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

Cf. A140711 (permutations by white "local" corners).

cp's OEIS Frontend

A140712 Number of white corners in all permutations of {1,2,...,n} (for definition see the Eriksson-Linusson references).

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A140713 Triangle read by rows: T(n,k) is the number of white corners of rank k in all the permutations of {1,2,...,n} (n>=2, 0<=k<=n-2; for definitions see the Eriksson-Linusson references).

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A213166 Triangle, read by rows, of permutations of length n with k white global corners.

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