Identifier
-
Mp00021:
Cores
—to bounded partition⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000246: Permutations ⟶ ℤ (values match St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.)
Values
([2],3) => [2] => [1,1,0,0,1,0] => [2,1,3] => 2
([1,1],3) => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2
([3,1],3) => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3
([2,1,1],3) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 3
([4,2],3) => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4
([3,1,1],3) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4
([2,2,1,1],3) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 4
([5,3,1],3) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([4,2,1,1],3) => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 5
([3,2,2,1,1],3) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 5
([6,4,2],3) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 6
([5,3,1,1],3) => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 6
([4,2,2,1,1],3) => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => 6
([3,3,2,2,1,1],3) => [1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,6,5,4,3,2] => 6
([2],4) => [2] => [1,1,0,0,1,0] => [2,1,3] => 2
([1,1],4) => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2
([3],4) => [3] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 3
([2,1],4) => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3
([1,1,1],4) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 3
([4,1],4) => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4
([2,2],4) => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4
([3,1,1],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4
([2,1,1,1],4) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 4
([5,2],4) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([4,1,1],4) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([3,2,1],4) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([3,1,1,1],4) => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 5
([2,2,1,1,1],4) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 5
([6,3],4) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 6
([5,2,1],4) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([4,1,1,1],4) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 6
([4,2,2],4) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 6
([3,3,1,1],4) => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 6
([3,2,1,1,1],4) => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => 6
([2,2,2,1,1,1],4) => [1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,6,5,4,3,2] => 6
([2],5) => [2] => [1,1,0,0,1,0] => [2,1,3] => 2
([1,1],5) => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2
([3],5) => [3] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 3
([2,1],5) => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3
([1,1,1],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 3
([4],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 4
([3,1],5) => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4
([2,2],5) => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4
([2,1,1],5) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4
([1,1,1,1],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 4
([5,1],5) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 5
([3,2],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([4,1,1],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([2,2,1],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([3,1,1,1],5) => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 5
([2,1,1,1,1],5) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 5
([6,2],5) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 6
([5,1,1],5) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 6
([3,3],5) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 6
([4,2,1],5) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([4,1,1,1],5) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 6
([2,2,2],5) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 6
([3,2,1,1],5) => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 6
([3,1,1,1,1],5) => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => 6
([2,2,1,1,1,1],5) => [1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,6,5,4,3,2] => 6
([2],6) => [2] => [1,1,0,0,1,0] => [2,1,3] => 2
([1,1],6) => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2
([3],6) => [3] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 3
([2,1],6) => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3
([1,1,1],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 3
([4],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 4
([3,1],6) => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4
([2,2],6) => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4
([2,1,1],6) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4
([1,1,1,1],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 4
([5],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 5
([4,1],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 5
([3,2],6) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([3,1,1],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([2,2,1],6) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([2,1,1,1],6) => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 5
([1,1,1,1,1],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 5
([6,1],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [4,5,3,2,1,6] => 6
([4,2],6) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 6
([5,1,1],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 6
([3,3],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 6
([3,2,1],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([4,1,1,1],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 6
([2,2,2],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 6
([2,2,1,1],6) => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 6
([3,1,1,1,1],6) => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => 6
([2,1,1,1,1,1],6) => [1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,6,5,4,3,2] => 6
([7,2],6) => [5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [4,3,5,2,1,6] => 7
([6,1,1],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [3,5,4,2,1,6] => 7
([4,3],6) => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 7
([5,2,1],6) => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 7
([5,1,1,1],6) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 7
([3,3,1],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 7
([3,2,2],6) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 7
([4,2,1,1],6) => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 7
([4,1,1,1,1],6) => [3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [1,5,4,6,3,2] => 7
([2,2,2,1],6) => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 7
([3,2,1,1,1],6) => [2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => 7
([3,1,1,1,1,1],6) => [2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,6,7,5,4,3,2] => 7
([2,2,1,1,1,1,1],6) => [1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,8,7,6,5,4,3,2] => 7
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Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Map
to 312-avoiding permutation
Description
Map
to bounded partition
Description
The (k-1)-bounded partition of a k-core.
Starting with a $k$-core, deleting all cells of hook length greater than or equal to $k$ yields a $(k-1)$-bounded partition [1, Theorem 7], see also [2, Section 1.2].
Starting with a $k$-core, deleting all cells of hook length greater than or equal to $k$ yields a $(k-1)$-bounded partition [1, Theorem 7], see also [2, Section 1.2].
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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