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In [[combinatorics|combinatorial mathematics]], '''Catalan's triangle''' is a [[number triangle]], where entry ''C<sub>n,k</sub>'' 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<ref>{{citeweb|url=http://oeis.org/A009766|author=The On-Line Encyclopedia of Integer Sequences|title=A009766: Catalan's triangle|accessdate=March 28, 2012}}</ref>
:{| class="wikitable"
|-
! n&nbsp;\&nbsp;k
! width="50" | 0
! width="50" | 1
! width="50" | 2
! width="50" | 3
! width="50" | 4
! width="50" | 5
! width="50" | 6
! width="50" | 7
! width="50" | 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
|}


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Each element is the sum of the one above and the one to the left. The diagonal ''C<sub>n,n</sub>'' consists of the [[Catalan numbers]].
 
== General formula ==
The general formula for ''C<sub>n,k</sub>'' is given by<ref>{{citeweb|url=http://mathworld.wolfram.com/CatalansTriangle.html|title=Catalan's Triangle|author=Eric W. Weisstein|location=MathWorld − A Wolfram Web Resource|accessdate=March 28, 2012}}</ref>
:<math>
C_{n,k} = \frac{(n+k)!(n-k+1)}{k!(n+1)!}. </math>
where ''n''! denotes the [[factorial]].
 
==See also==
*[[Pascal's triangle]]
 
== References ==
{{Reflist}}
 
[[Category:Triangles of numbers]]

Revision as of 23:47, 16 February 2013

In combinatorial mathematics, Catalan's triangle is a number triangle, where entry Cn,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[1]

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 Cn,n consists of the Catalan numbers.

General formula

The general formula for Cn,k is given by[2]

Cn,k=(n+k)!(n−k+1)k!(n+1)!.

where n! denotes the factorial.

See also

References

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  1. ↑ Template:Citeweb
  2. ↑ Template:Citeweb
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