Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 FussCatalanNumber

Source Notebook

Evaluate the Fuss–Catalan number

Contributed by: Jan Mangaldan

ResourceFunction["FussCatalanNumber"][n,m]

gives the Fuss–Catalan number .

Details

Integer mathematical function, suitable for both symbolic and numerical manipulation.
ResourceFunction["FussCatalanNumber"][n,m] is generically defined as .
Fuss–Catalan numbers are integers for integer arguments and appear in various tree- and path-enumeration problems.
ResourceFunction["FussCatalanNumber"] automatically threads over lists.

Examples

Basic Examples (1) 

Evaluate numerically:

In[1]:=
ResourceFunction["FussCatalanNumber"][5, 3]
Out[1]=

Scope (3) 

Evaluate for large arguments:

In[2]:=
ResourceFunction["FussCatalanNumber"][1000, 100] // Short
Out[2]=

Evaluate for half-integer arguments:

In[3]:=
ResourceFunction["FussCatalanNumber"][3/2, 6]
Out[3]=
In[4]:=
ResourceFunction["FussCatalanNumber"][7, 3/2]
Out[4]=

FussCatalanNumber threads elementwise over lists:

In[5]:=
ResourceFunction["FussCatalanNumber"][{1, 2, 3, 4}, 5]
Out[5]=

Applications (2) 

FussCatalanNumber[n,m] counts the number of groupings of 1,…,(m-1)n+1 taken m at a time:

In[6]:=
With[{n = 5, m = 7}, Length[Groupings[(m - 1) n + 1, m]]]
Out[6]=
In[7]:=
ResourceFunction["FussCatalanNumber"][5, 7]
Out[7]=

Use FussCatalanNumber to count the number of m-Raney sequences:

In[8]:=
With[{n = 6, m = 3}, Count[Tuples[{1, 1 - m}, n m + 1], v_ /; Total[v] == 1 && VectorQ[Accumulate[v], Positive]]]
Out[8]=
In[9]:=
ResourceFunction["FussCatalanNumber"][6, 3]
Out[9]=

Properties and Relations (2) 

FussCatalanNumber[n,2] is equivalent to CatalanNumber[n]:

In[10]:=
ResourceFunction["FussCatalanNumber"][n, 2] == CatalanNumber[n] // FullSimplify
Out[10]=

A convolution relation satisfied by FussCatalanNumber:

In[11]:=
With[{n = 9, m = 3}, ResourceFunction["FussCatalanNumber"][n, m] == Boole[n == 0] + Sum[Product[ ResourceFunction["FussCatalanNumber"][k, m], {k, id}], {id, FrobeniusSolve[ConstantArray[1, m], n - 1]}]]
Out[11]=

Version History

  • 1.0.0 – 28 June 2021

Source Metadata

  • Citation:
    • Graham, R.L., Knuth, D.E., Patashnik, O., Concrete Mathematics: A Foundation for Computer Science, Second Edition. Reading, MA: Addison-Wesley, p. 361, 1994.

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