Integer partitions, the partition function, and Young diagrams

Enumerates every partition of 6, checks the count against the partition function p(6), and visualizes one partition’s Young diagram and its conjugate.

from mathematicskit.combinatorics import YoungDiagram, integer_partitions, partition_function
from mathematicskit.combinatorics.visualizers.plots import plot_partition_counts, plot_young_diagram

Enumerate the partitions of 6

partitions = integer_partitions(6)
print(f"p(6) = {partition_function(6)}, enumerated {len(partitions)} partitions:")
for p in partitions:
    print(" ", p)

p(6) = 11, enumerated 11 partitions:
  [6]
  [5, 1]
  [4, 2]
  [4, 1, 1]
  [3, 3]
  [3, 2, 1]
  [3, 1, 1, 1]
  [2, 2, 2]
  [2, 2, 1, 1]
  [2, 1, 1, 1, 1]
  [1, 1, 1, 1, 1, 1]

A Young diagram and its conjugate

diagram = YoungDiagram([4, 2, 1])
print("\nFerrers diagram of (4, 2, 1):")
print(diagram.ferrers_diagram())
print("conjugate partition:", diagram.conjugate().parts)

plot_young_diagram(diagram)

Ferrers diagram of (4, 2, 1):
****
**
*
conjugate partition: [3, 2, 1, 1]

<Axes: title={'center': 'Young diagram of [4, 2, 1]'}>

Growth of the partition function

plot_partition_counts(30)

<Axes: title={'center': 'Integer partition function'}, xlabel='n', ylabel='p(n)'>