A285852 Rectangular array read by rows: T(n,k) is the number of words of length n on alphabet {0,1,2} that have exactly k records, n>=0, 0<=k<=3.

1, 0, 0, 0, 0, 3, 0, 0, 0, 6, 3, 0, 0, 14, 12, 1, 0, 36, 39, 6, 0, 98, 120, 25, 0, 276, 363, 90, 0, 794, 1092, 301, 0, 2316, 3279, 966, 0, 6818, 9840, 3025, 0, 20196, 29523, 9330, 0, 60074, 88572, 28501, 0, 179196, 265719, 86526

Offset: 0

Author

Geoffrey Critzer, Apr 27 2017

Keywords

• nonn
• tabf

Comments

A record in a word a_1,a_2,...,a_n is a letter a_j that is larger than all the preceding letters. That is, a_j>a_i for all i

Examples

			1,    0,    0,   0;
0,    3,    0,   0;
0,    6,    3,   0;
0,   14,   12,   1;
0,   36,   39,   6;
0,   98,  120,  25;
0,  276,  363,  90;
0,  794, 1092, 301;
0, 2316, 3279, 966;
		

Links

• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009.

Crossrefs

Column k=0 gives A000007.

Column k=1 gives A001550.

Column k=2 gives A029858.

Column k=3 gives A000392.

Row sums give A000244.

Programs

Mathematica

nn = 12;CoefficientList[Series[Product[1 + u z/(1 - j z), {j, 1, 3}], {z, 0, nn}], {z,u}] // Grid

Formula

G.f.: Product_{j=1..3} (1 + y*x/(1 - j*x)). Generally for words on alphabet {0,1,...,r} the o.g.f. is Product_{j=1..r} (1 + y*x/(1 - j*x)).

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