A124774 Multinomial coefficients for compositions in standard order.

1, 1, 1, 2, 1, 3, 3, 6, 1, 4, 6, 12, 4, 12, 12, 24, 1, 5, 10, 20, 10, 30, 30, 60, 5, 20, 30, 60, 20, 60, 60, 120, 1, 6, 15, 30, 20, 60, 60, 120, 15, 60, 90, 180, 60, 180, 180, 360, 6, 30, 60, 120, 60, 180, 180, 360, 30, 120, 180, 360, 120, 360, 360, 720, 1, 7, 21, 42, 35, 105

Offset: 0

Author

Franklin T. Adams-Watters, Nov 06 2006

Keywords

• easy
• nonn
• tabf

Comments

The standard order of compositions is given by A066099.

Number of ways to distribute labeled objects into boxes, with the number of objects in each box being specified by the composition.

Examples

			Composition number 11 is 2,1,1; there are 6 choices for the pair of objects in the first box, then 2 choices for the object in the next box, so a(11) = 6*2 = 12.
The table starts:
1
1
1 2
1 3 3 6
		

Links

• Alois P. Heinz, Rows n = 0..14, flattened
• Thomas Garrity and Jacob Lehmann Duke, Ergodicity and Algebraticity of the Fast and Slow Triangle Maps, arXiv:2409.05822 [math.DS], 2024. See p. 22.

Crossrefs

Cf. A066099, A124773, A011782 (row lengths), A000670 (row sums), A036039.

Formula

For composition b(1),...,b(k), a(n) = (Sum_{i=1}^k b(i))! / (Product_{i=1}^k b(i)!).