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Catalan's triangle

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In combinatorial mathematics, Catalan's triangle is a number triangle whose entries C ( n , k ) give 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. Bailey shows that C ( n , k ) satisfy the following properties:

Contents

  1. C ( n , 0 ) = 1  for  n ≥ 0 .
  2. C ( n , 1 ) = n  for  n ≥ 1 .
  3. C ( n + 1 , k ) = C ( n + 1 , k − 1 ) + C ( n , k )  for  1 < k < n + 1
  4. C ( n + 1 , n + 1 ) = C ( n + 1 , n )  for  n ≥ 1 .

Formula 3 shows that the entry in the triangle is obtained recursively by adding numbers to the left and above in the triangle. The earliest appearance of the Catalan triangle along with the recursion formula is in page 214 of the treatise on Calculus published in 1800 by Louis François Antoine Arbogast.

Shapiro introduces another triangle which he calls the Catalan triangle that is distinct from the triangle being discussed here.

General formula

The general formula for C ( n , k ) is given by

C ( n , k ) = ( n + k ) ! ( n − k + 1 ) k ! ( n + 1 ) ! ,

where n and k are nonnegative integers and n! denotes the factorial. Thus,, for k>0, C ( n , k ) = ( n + k k ) − ( n + k k − 1 ) .

The diagonal C(n, n) is the n-th Catalan number. The row sum of the n-th row is the (n + 1)-th Catalan number.

Some values are given by

Generalization

Catalan's trapezoids are a countable set of number trapezoids which generalize Catalan’s triangle. Catalan's trapezoid of order m = 1, 2, 3, ... is a number trapezoid whose entries C m ( n , k ) give the number of strings consisting of n X-s and k Y-s such that in every initial segment of the string the number of Y-s does not exceed the number of X-s by m or more. By definition, Catalan's trapezoid of order m = 1 is Catalan's triangle, i.e., C 1 ( n , k ) = C ( n , k ) .

Some values of Catalan's trapezoid of order m = 2 are given by

Some values of Catalan's trapezoid of order m = 3 are given by

Again, each element is the sum of the one above and the one to the left.

A general formula for C m ( n , k ) is given by

C m ( n , k ) = { ( n + k k ) 0 ≤ k < m ( n + k k ) − ( n + k k − m ) m ≤ k ≤ n + m − 1 0 k > n + m − 1

( n = 0, 1, 2, ..., k = 0, 1, 2, ..., m = 1, 2, 3, ...).

References

Catalan's triangle Wikipedia
(Text) CC BY-SA


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