Young’s lattice is the infinite lattice of all partitions with partial ordering given by componentwise linear ordering.
i1 : youngSubposet 4 o1 = Poset{cache => CacheTable{...8...} } GroundSet => {{}, {1}, {2}, {1, 1}, {3}, {2, 1}, {1, 1, 1}, {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}} RelationMatrix => | 1 1 1 1 1 1 1 1 1 1 1 1 | | 0 1 1 1 1 1 1 1 1 1 1 1 | | 0 0 1 0 1 1 0 1 1 1 1 0 | | 0 0 0 1 0 1 1 0 1 1 1 1 | | 0 0 0 0 1 0 0 1 1 0 0 0 | | 0 0 0 0 0 1 0 0 1 1 1 0 | | 0 0 0 0 0 0 1 0 0 0 1 1 | | 0 0 0 0 0 0 0 1 0 0 0 0 | | 0 0 0 0 0 0 0 0 1 0 0 0 | | 0 0 0 0 0 0 0 0 0 1 0 0 | | 0 0 0 0 0 0 0 0 0 0 1 0 | | 0 0 0 0 0 0 0 0 0 0 0 1 | Relations => {{{}, {1}}, {{1}, {2}}, {{1}, {1, 1}}, {{2}, {3}}, {{2}, {2, 1}}, {{1, 1}, {2, 1}}, {{1, 1}, {1, 1, 1}}, {{3}, {4}}, {{3}, {3, 1}}, {{2, 1}, {3, 1}}, {{2, 1}, {2, 2}}, {{2, 1}, {2, 1, 1}}, {{1, 1, 1}, {2, 1, 1}}, {{1, 1, 1}, {1, 1, 1, 1}}} o1 : Poset |
i2 : youngSubposet({3,1}, {4,2,1}) o2 = Poset{cache => CacheTable{} } GroundSet => {{3, 1}, {3, 1, 1}, {3, 2}, {3, 2, 1}, {4, 1}, {4, 1, 1}, {4, 2}, {4, 2, 1}} RelationMatrix => | 1 1 1 1 1 1 1 1 | | 0 1 0 1 0 1 0 1 | | 0 0 1 1 0 0 1 1 | | 0 0 0 1 0 0 0 1 | | 0 0 0 0 1 1 1 1 | | 0 0 0 0 0 1 0 1 | | 0 0 0 0 0 0 1 1 | | 0 0 0 0 0 0 0 1 | Relations => {{{3, 1}, {3, 1, 1}}, {{3, 1}, {3, 2}}, {{3, 1}, {3, 2, 1}}, {{3, 1}, {4, 1}}, {{3, 1}, {4, 1, 1}}, {{3, 1}, {4, 2}}, {{3, 1}, {4, 2, 1}}, {{3, 1, 1}, {3, 2, 1}}, {{3, 1, 1}, {4, 1, 1}}, {{3, 1, 1}, {4, 2, 1}}, {{3, 2}, {3, 2, 1}}, {{3, 2}, {4, 2}}, {{3, 2}, {4, 2, 1}}, {{3, 2, 1}, {4, 2, 1}}, {{4, 1}, {4, 1, 1}}, {{4, 1}, {4, 2}}, {{4, 1}, {4, 2, 1}}, {{4, 1, 1}, {4, 2, 1}}, {{4, 2}, {4, 2, 1}}} o2 : Poset |