A212850 Number of n X 3 arrays with rows being permutations of 0..2 and no column j greater than column j-1 in all rows.

1, 19, 163, 1135, 7291, 45199, 275563, 1666495, 10038331, 60348079, 362442763, 2175719455, 13057505371, 78354598159, 470156286763, 2821023814015, 16926401164411, 101559181827439, 609357415487563, 3656151466494175

Offset: 1

Author

R. H. Hardin, May 28 2012

Keywords

• nonn

Comments

From Petros Hadjicostas, Aug 25 2019: (Start)

Both formulas below follow from the theory in the documentation of array A309951. We have Sum_{s = 0..A000041(3)} (-1)^s * A309951(3,s) * a(n-s) = 0, i.e., a(n) - 10*a(n-1) - 27*a(n-2) + 18*a(n-3) = 0 for n >= 4. This is a consequence of Eq. (6) on p. 248 of Abramson and Promislow (1978), where we let t=0 in the equation.

In the explicit formula by Vaclav Kotesovec below, a(n) = 6^n - 2*3^n + 1^n, the numbers 1, 3, 6 (that are raised to the n-th power) are the multinomial coefficients of the A000041(3) = 3 integer partitions of 3: 1 = 3!/3!, 3 = 3!/(1!2!), 6 = 3!/(1!1!1!).

(End)

Examples

			Some solutions for n=3:
  1 2 0   2 1 0   0 2 1   1 2 0   1 2 0   2 1 0   1 2 0
  2 0 1   2 0 1   2 0 1   2 0 1   0 2 1   2 0 1   1 0 2
  0 2 1   0 1 2   2 1 0   2 1 0   2 0 1   0 2 1   0 2 1
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210
• Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (6) on p. 248 (set t:=0).

Crossrefs

Column 3 of A212855.

Cf. A000041, A212851, A212852, A212852, A212853, A212854, A212856, A309951.

Formula

Empirical: a(n) = 10*a(n-1) - 27*a(n-2) + 18*a(n-3).

Explicit formula: a(n) = 6^n - 2*3^n + 1. - Vaclav Kotesovec, May 31 2012