- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000339: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{(2, 2)} = 2 + q$

$F_{(2, 3)} = 3 + 2\ q$

$F_{(3, 2)} = 3 + 2\ q$

$F_{(2, 4)} = 4 + 3\ q$

$F_{(3, 3)} = 6 + 6\ q + q^{2}$

$F_{(4, 2)} = 4 + q^{2} + 2\ q^{3} + q^{4}$

$F_{(2, 5)} = 5 + 4\ q$

$F_{(3, 4)} = 10 + 12\ q + 3\ q^{2}$

$F_{(4, 3)} = 10 + 6\ q^{2} + 9\ q^{3} + 5\ q^{4}$

$F_{(5, 2)} = 5 + q^{2} + q^{3} + q^{4} + 2\ q^{5} + q^{8}$

$F_{(2, 6)} = 6 + 5\ q$

$F_{(3, 5)} = 15 + 20\ q + 6\ q^{2}$

$F_{(4, 4)} = 20 + 18\ q^{2} + 25\ q^{3} + 14\ q^{4}$

$F_{(5, 3)} = 15 + 5\ q^{2} + 6\ q^{3} + 9\ q^{4} + 12\ q^{5} + 3\ q^{6} + q^{7} + 6\ q^{8}$

$F_{(6, 2)} = 6 + q^{3} + 2\ q^{4} + q^{6} + 3\ q^{7} + q^{8} + q^{13}$

$F_{(2, 7)} = 7 + 6\ q$

$F_{(3, 6)} = 21 + 30\ q + 10\ q^{2}$

$F_{(4, 5)} = 35 + 40\ q^{2} + 54\ q^{3} + 30\ q^{4}$

$F_{(5, 4)} = 35 + 15\ q^{2} + 21\ q^{3} + 35\ q^{4} + 41\ q^{5} + 17\ q^{6} + 5\ q^{7} + 21\ q^{8} + q^{9}$

$F_{(6, 3)} = 21 + 7\ q^{3} + 14\ q^{4} + 4\ q^{5} + 11\ q^{6} + 21\ q^{7} + 10\ q^{8} + 5\ q^{9} + 2\ q^{10} + q^{12} + 7\ q^{13}$

$F_{(2, 8)} = 8 + 7\ q$

$F_{(3, 7)} = 28 + 42\ q + 15\ q^{2}$

$F_{(4, 6)} = 56 + 75\ q^{2} + 100\ q^{3} + 55\ q^{4}$

$F_{(5, 5)} = 70 + 35\ q^{2} + 56\ q^{3} + 95\ q^{4} + 105\ q^{5} + 55\ q^{6} + 15\ q^{7} + 55\ q^{8} + 5\ q^{9}$

$F_{(6, 4)} = 56 + 27\ q^{3} + 54\ q^{4} + 26\ q^{5} + 54\ q^{6} + 89\ q^{7} + 48\ q^{8} + 35\ q^{9} + 15\ q^{10} + q^{11} + 9\ q^{12} + 27\ q^{13}$

Description

The maf index of a permutation.
Let

$\sigma$ be a permutation with fixed point set

$\operatorname{FIX}(\sigma)$ , and let

$\operatorname{Der}(\sigma)$ be the derangement obtained from

$\sigma$ by removing the fixed points.
Then

$\operatorname{maf}(\sigma) = \sum_{i \in \operatorname{FIX}(\sigma)} i - \binom{|\operatorname{FIX}(\sigma)|+1}{2} + \operatorname{maj}(\operatorname{Der}(\sigma)),$
where

$\operatorname{maj}(\operatorname{Der}(\sigma))$ is the major index of the derangement of

$\sigma$ .

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).

Map

ones to leading

Description

The unique permutation obtained by applying the Foata-Riordan map to obtain a Prüfer code, then prepending zero and cyclically shifting.
Let

$c_1,\dots, c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let

$c_0 = 0$ .
This map returns the a unique permutation

$q_1,\dots, q_n$ such that

$q_i - c_{i-1}$ is constant modulo

$n+1$ .
This map is Mp00299ones to leading restricted to permutations.