A062154 Number T(n,m) of n X m matrices over {0,1,2} with all row and column sums equal to 1 or 2, m=0,..,2*n.
1, 0, 2, 1, 0, 1, 13, 18, 6, 0, 0, 18, 189, 450, 360, 90, 0, 0, 6, 450, 4842, 16380, 22140, 12600, 2520, 0, 0, 0, 360, 16380, 190080, 832950, 1631700, 1537200, 680400, 113400, 0, 0, 0, 90, 22140, 832950, 10520010, 56609280, 147533400, 200377800
Offset: 0
Examples
Triangle begins:
[0] 1;
[1] 0, 2, 1;
[2] 0, 1, 13, 18, 6;
[3] 0, 0, 18, 189, 450, 360, 90;
[4] 0, 0, 6, 450, 4842, 16380, 22140, 12600, 2520;
[5] 0, 0, 0, 360, 16380, 190080, 832950, 1631700, 1537200, 680400, 113400;
[6] 0, 0, 0, 90, 22140, 832950, 10520010, 56609280, 147533400, 200377800, 144585000, 52390800, 7484400;
T(2, 2)=13, i.e. there are 13 2 X 2 matrices over {0, 1, 2} with all row and column sums equal to 1 or 2: [0 1 / 0 1], [0 1 / 0 2], [0 2 / 1 0], [1 0 / 1 0], [1 1 / 1 1], [1 1 / 2 0], [2 0 / 1 0], [1 1 / 2 0], [1 0 / 2 0], [0 1 / 0 2], [1 1 / 0 1], [1 0 / 1 1], [0 1 / 0 2].
References
- I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(Problem 3.4.15).
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..960 (rows 0..30)
Programs
-
PARI
Row(n)={Vecrev(serlaplace(n!*polcoef((1/sqrt(1-x*y + O(x*x^n))*exp(x*y/2+1/(1-x*y)*(x*y+x^2*y/2+x*y^2/2) + O(x*x^n))), n)))} { for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Feb 03 2021
Formula
Sum_{n >= 0, m >= 0} T(n, m)*x^n/n!*y^m/m! = 1/sqrt(1-x*y)*exp(x*y/2+1/(1-x*y)*(x*y+x^2*y/2+x*y^2/2)).
Sum_{n >= 0, m >= 0} T(n, m)*x^n/n!*y^m/m! = 1+(1/2*y^2+2*y)*x+(1/8*y^4+3/2*y^3+13/4*y^2+1/2*y)*x^2+(1/48*y^6+1/2*y^5+25/8*y^4+21/4*y^3+3/2*y^2)*x^3+...