Negativity (quantum mechanics)

In combinatorial mathematics, Catalan's triangle is a number triangle, where entry C_n,k denotes the number of strings consisting of n X's and k Y's such that no initial segment of the string has more Y's than X's. It is a generalization of the Catalan numbers, and is named after Eugène Charles Catalan.

Some values are given by

n \ k	0	1	2	3	4	5	6	7	8
0	1
1	1	1
2	1	2	2
3	1	3	5	5
4	1	4	9	14	14
5	1	5	14	28	42	42
6	1	6	20	48	90	132	132
7	1	7	27	75	165	297	429	429
8	1	8	35	110	275	572	1001	1430	1430

Each element is the sum of the one above and the one to the left. The diagonal C_n,n consists of the Catalan numbers.

General formula

The general formula for C_n,k is given by

$C_{n,k}=\frac{(n+k)!(n-k+1)}{k!(n+1)!}.$

where n! denotes the factorial.

See also

• Pascal's triangle

References

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