A323846 Array read by antidiagonals: T(m,n) = number of m X n matrices M with entries {0,1,2} that have M_{1,1}=0, M_{m,n}=2, are such that the rows and columns are monotonic without jumps of 2, and satisfy M_{(i+1),(j+1)} = M_{i,j} + (0 or 1).

0, 0, 0, 1, 0, 1, 3, 4, 4, 3, 6, 16, 25, 16, 6, 10, 41, 94, 94, 41, 10, 15, 85, 266, 386, 266, 85, 15, 21, 155, 632, 1247, 1247, 632, 155, 21, 28, 259, 1332, 3423, 4657, 3423, 1332, 259, 28, 36, 406, 2570, 8342, 14795, 14795, 8342, 2570, 406, 36, 45, 606, 4631, 18546, 41586, 54219, 41586, 18546, 4631, 606, 45

Offset: 1

Author

N. J. A. Sloane, Feb 06 2019

Keywords

• nonn
• tabl

Comments

The monotonicity condition requires that M_{(i+1),j} = M_{i,j} + (0 or 1); M_{i,(j+1)} = M_{i,j} + (0 or 1).

These matrices can be cut into three connected pieces, containing the 0's, 1's, and 2's; there are two vertex-disjoint paths from the north-and-east edges of the matrix to the south-and-west edges.

Row (or column) n >= 1 has a linear recurrence (with constant coefficients) of order 2n+1. - Alois P. Heinz, Feb 07 2019

Examples

			Array begins:
    0   0    1    3     6    10 ...
    0   0    4   16    41    85 ...
    1   4   25   94   266   632 ...
    3  16   94  386  1247  3423 ...
    6  41  266 1247  4657 14795 ...
   10  85  632 3427 14795 54219 ...
...
The 4 examples when m=2 and n=3 are
    011   011  012   012
    012   112  012   112
		

References

• D. E. Knuth, Email to N. J. A. Sloane, Feb 05 2019.

Links

• Alois P. Heinz, Antidiagonals n = 1..80, flattened

Crossrefs

Rows 1-10 give: A000217(n-2), A323847, A323967, A323968, A323969, A323970, A323971, A323972, A323973, A323974.

Main diagonal gives A306322.

Cf. A132823, A252876, A229428.

Extensions

More terms from Alois P. Heinz, Feb 07 2019