- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000055: Permutations ⟶ ℤ (values match St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,1],[2,2]] => [3,4,1,2] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,1],[2,3]] => [3,4,1,2] => 8

[[1,1],[3,3]] => [3,4,1,2] => 8

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

[[1,2],[3,3]] => [3,4,1,2] => 8

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Description

The inversion sum of a permutation.
A pair $a < b$ is an inversion of a permutation $\pi$ if $\pi(a) > \pi(b)$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$.
This is also half of the metric associated with Spearmans coefficient of association $\rho$, $\sum_i (\pi_i - i)^2$, see [5].
This is also equal to the total number of occurrences of the classical permutation patterns $[2,1], [2, 3, 1], [3, 1, 2]$, and $[3, 2, 1]$, see [2].
This is also equal to the rank of the permutation inside the alternating sign matrix lattice, see references [2] and [3].
This lattice is the MacNeille completion of the strong Bruhat order on the symmetric group [1], which means it is the smallest lattice containing the Bruhat order as a subposet. This is a distributive lattice, so the rank of each element is given by the cardinality of the associated order ideal. The rank is calculated by summing the entries of the corresponding monotone triangle and subtracting $\binom{n+2}{3}$, which is the sum of the entries of the monotone triangle corresponding to the identity permutation of $n$.
This is also the number of bigrassmannian permutations (that is, permutations with exactly one left descent and one right descent) below a given permutation $\pi$ in Bruhat order, see Theorem 1 of [6].

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).