Elder's Theorem

Author

Eric W. Weisstein
April 10, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/EldersTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/EldersTheorem.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<DiscreteMath`Combinatorica`

Count the Number of Times That k Occurs

Table[Split[Sort[Flatten[Partitions[n]]]],{n,5}]

TableForm[Table[Length/@Split[Sort[Flatten[Partitions[n]]]],{n,10}],TableSpacing->1,TableAlignments->Right]

Which is equivalent to

ColumnForm[cnts=Table[Count[#,k]&/@Partitions[n],{n,5},{k,n}]
]

Count the Number of Occasions That a Part Occurs k or More Times

Table[Table[Count[Flatten[Split/@Partitions[n],1],_?(Length[#]≥k&)],
{k,n}],{n,5}]

Example for n=4

Partitions[4]

Split/@Partitions[4]