- FindStat

Identifier

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing 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

[[1],[]] => [] => [] => [] => 0

[[2],[]] => [] => [] => [] => 0

[[1,1],[]] => [] => [] => [] => 0

[[2,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[3],[]] => [] => [] => [] => 0

[[2,1],[]] => [] => [] => [] => 0

[[3,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[2,2],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[3,2],[2]] => [2] => [1,1,0,0,1,0] => [2,1,3] => 2

[[1,1,1],[]] => [] => [] => [] => 0

[[2,2,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2

[[2,1,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[3,2,1],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

[[4],[]] => [] => [] => [] => 0

[[3,1],[]] => [] => [] => [] => 0

[[4,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[2,2],[]] => [] => [] => [] => 0

[[3,2],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[4,2],[2]] => [2] => [1,1,0,0,1,0] => [2,1,3] => 2

[[2,1,1],[]] => [] => [] => [] => 0

[[3,2,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2

[[3,1,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[4,2,1],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

[[3,3],[2]] => [2] => [1,1,0,0,1,0] => [2,1,3] => 2

[[4,3],[3]] => [3] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 3

[[2,2,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[3,3,1],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

[[3,2,1],[2]] => [2] => [1,1,0,0,1,0] => [2,1,3] => 2

[[4,3,1],[3,1]] => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4

[[2,2,2],[1,1]] => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2

[[3,3,2],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4

[[3,2,2],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

[[4,3,2],[3,2]] => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5

[[1,1,1,1],[]] => [] => [] => [] => 0

[[2,2,2,1],[1,1,1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 3

[[2,2,1,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2

[[3,3,2,1],[2,2,1]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5

[[2,1,1,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[3,2,2,1],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4

[[3,2,1,1],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

[[4,3,2,1],[3,2,1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6

[[5],[]] => [] => [] => [] => 0

[[4,1],[]] => [] => [] => [] => 0

[[5,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[3,2],[]] => [] => [] => [] => 0

[[4,2],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[5,2],[2]] => [2] => [1,1,0,0,1,0] => [2,1,3] => 2

[[3,1,1],[]] => [] => [] => [] => 0

[[4,2,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2

[[4,1,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[5,2,1],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

[[3,3],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[4,3],[2]] => [2] => [1,1,0,0,1,0] => [2,1,3] => 2

[[5,3],[3]] => [3] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 3

[[2,2,1],[]] => [] => [] => [] => 0

[[3,3,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2

[[3,2,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[4,3,1],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

[[4,2,1],[2]] => [2] => [1,1,0,0,1,0] => [2,1,3] => 2

[[5,3,1],[3,1]] => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4

[[3,2,2],[1,1]] => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2

[[4,3,2],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4

[[4,2,2],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

[[5,3,2],[3,2]] => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5

[[2,1,1,1],[]] => [] => [] => [] => 0

[[3,2,2,1],[1,1,1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 3

[[3,2,1,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2

[[4,3,2,1],[2,2,1]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5

[[3,1,1,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[4,2,2,1],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4

[[4,2,1,1],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

[[5,3,2,1],[3,2,1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6

[[4,4],[3]] => [3] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 3

[[5,4],[4]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 4

[[3,3,1],[2]] => [2] => [1,1,0,0,1,0] => [2,1,3] => 2

[[4,4,1],[3,1]] => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4

[[4,3,1],[3]] => [3] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 3

[[5,4,1],[4,1]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => 5

[[2,2,2],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[3,3,2],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

[[4,4,2],[3,2]] => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5

[[3,2,2],[2]] => [2] => [1,1,0,0,1,0] => [2,1,3] => 2

[[4,3,2],[3,1]] => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4

[[5,4,2],[4,2]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 6

[[2,2,1,1],[1]] => [1] => [1,0,1,0] => [1,2] => 1

[[3,3,2,1],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4

[[3,3,1,1],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

[[4,4,2,1],[3,2,1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6

[[3,2,1,1],[2]] => [2] => [1,1,0,0,1,0] => [2,1,3] => 2

[[4,3,2,1],[3,1,1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5

[[4,3,1,1],[3,1]] => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4

[[5,4,2,1],[4,2,1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 7

[[3,3,3],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4

[[4,4,3],[3,3]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 6

[[4,3,3],[3,2]] => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5

[[5,4,3],[4,3]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 7

[[2,2,2,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2

[[3,3,3,1],[2,2,1]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5

[[3,3,2,1],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4

[[4,4,3,1],[3,3,1]] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 7

[[3,2,2,1],[2,1]] => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3

>>> Load all 1651 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 2 + q$

$F_{3} = 3 + 3\ q + 2\ q^{2} + q^{3}$

$F_{4} = 5 + 5\ q + 6\ q^{2} + 6\ q^{3} + 3\ q^{4} + 2\ q^{5} + q^{6}$

$F_{5} = 7 + 9\ q + 12\ q^{2} + 14\ q^{3} + 13\ q^{4} + 11\ q^{5} + 9\ q^{6} + 6\ q^{7} + 3\ q^{8} + 2\ q^{9} + q^{10}$

$F_{6} = 11 + 13\ q + 22\ q^{2} + 27\ q^{3} + 31\ q^{4} + 33\ q^{5} + 30\ q^{6} + 28\ q^{7} + 23\ q^{8} + 19\ q^{9} + 14\ q^{10} + 9\ q^{11} + 6\ q^{12} + 3\ q^{13} + 2\ q^{14} + q^{15}$

Description

The number of non-inversions of a permutation.
For a permutation of

$\{1,\ldots,n\}$ , this is given by

$\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$ .

Map

to non-crossing permutation

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

Map

inner shape

Description

The inner shape of a skew partition.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.