- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001759: Permutations ⟶ ℤ

Values

{{1}} => [1] => [1] => [1] => 0

{{1,2}} => [2,1] => [2,1] => [2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,5},{4}} => [2,3,5,4,1] => [5,2,3,4,1] => [5,1,2,4,3] => 7

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

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

{{1,2,4,5},{3}} => [2,4,3,5,1] => [4,2,3,5,1] => [5,1,3,2,4] => 6

{{1,2,4},{3,5}} => [2,4,5,1,3] => [4,5,2,1,3] => [4,1,5,2,3] => 6

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

{{1,2,5},{3,4}} => [2,5,4,3,1] => [5,4,2,3,1] => [5,1,4,3,2] => 9

{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,4,5,1,3] => [1,3,5,2,4] => 6

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

{{1,2,5},{3},{4}} => [2,5,3,4,1] => [2,5,3,4,1] => [5,2,1,4,3] => 8

{{1,2},{3,5},{4}} => [2,1,5,4,3] => [5,2,4,1,3] => [1,3,5,4,2] => 7

{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,5,1,3,4] => [1,3,2,5,4] => 6

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

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

{{1,3,4},{2,5}} => [3,5,4,1,2] => [5,3,1,4,2] => [4,5,1,3,2] => 8

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

{{1,3,5},{2,4}} => [3,4,5,2,1] => [3,4,5,2,1] => [5,4,1,2,3] => 7

{{1,3},{2,4,5}} => [3,4,1,5,2] => [3,1,4,5,2] => [3,5,1,2,4] => 6

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

{{1,3,5},{2},{4}} => [3,2,5,4,1] => [3,5,2,4,1] => [5,1,3,4,2] => 7

{{1,3},{2,5},{4}} => [3,5,1,4,2] => [3,5,1,4,2] => [2,5,3,4,1] => 7

{{1,3},{2},{4,5}} => [3,2,1,5,4] => [3,2,5,1,4] => [1,4,3,5,2] => 6

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

{{1,4,5},{2,3}} => [4,3,2,5,1] => [4,3,2,5,1] => [5,3,2,1,4] => 7

{{1,4},{2,3,5}} => [4,3,5,1,2] => [4,3,1,5,2] => [4,5,2,1,3] => 7

{{1,4},{2,3},{5}} => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 6

{{1,5},{2,3,4}} => [5,3,4,2,1] => [3,5,4,2,1] => [5,4,2,3,1] => 9

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

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

{{1,5},{2,3},{4}} => [5,3,2,4,1] => [3,2,5,4,1] => [5,3,2,4,1] => 8

{{1},{2,3,5},{4}} => [1,3,5,4,2] => [5,3,4,1,2] => [1,5,2,4,3] => 7

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

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

{{1,4,5},{2},{3}} => [4,2,3,5,1] => [2,4,3,5,1] => [5,2,3,1,4] => 6

{{1,4},{2,5},{3}} => [4,5,3,1,2] => [4,1,5,3,2] => [4,5,3,1,2] => 8

{{1,4},{2},{3,5}} => [4,2,5,1,3] => [2,4,1,5,3] => [4,2,5,1,3] => 7

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

{{1,5},{2,4},{3}} => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 10

{{1},{2,4,5},{3}} => [1,4,3,5,2] => [4,3,5,1,2] => [1,5,3,2,4] => 6

{{1},{2,4},{3,5}} => [1,4,5,2,3] => [4,1,5,2,3] => [3,1,5,2,4] => 6

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

{{1,5},{2},{3,4}} => [5,2,4,3,1] => [5,2,4,3,1] => [5,2,4,3,1] => 9

{{1},{2,5},{3,4}} => [1,5,4,3,2] => [5,4,3,1,2] => [1,5,4,3,2] => 9

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

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

{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [2,3,5,4,1] => [5,2,3,4,1] => 7

{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [3,5,4,1,2] => [1,5,3,4,2] => 7

{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [5,4,1,2,3] => [1,2,5,4,3] => 7

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

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

{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 5

{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [2,3,4,5,1,6] => [5,1,2,3,4,6] => 4

{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [6,2,3,4,5,1] => [6,1,2,3,5,4] => 9

{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [2,3,4,6,1,5] => [1,3,4,5,6,2] => 5

{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [2,3,4,1,5,6] => [4,1,2,3,5,6] => 3

{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [5,2,3,4,6,1] => [6,1,2,4,3,5] => 8

{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [5,6,2,3,1,4] => [5,1,2,6,3,4] => 8

{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [5,2,3,4,1,6] => [5,1,2,4,3,6] => 7

{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [6,5,2,3,4,1] => [6,1,2,5,4,3] => 12

{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [2,3,5,6,1,4] => [1,3,4,6,2,5] => 8

{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [2,3,5,1,4,6] => [1,3,4,5,2,6] => 4

{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [2,6,3,4,5,1] => [6,2,1,3,5,4] => 10

{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [6,2,3,5,1,4] => [1,3,4,6,5,2] => 9

{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [2,3,6,1,4,5] => [1,3,4,2,6,5] => 8

{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [2,3,1,4,5,6] => [3,1,2,4,5,6] => 2

{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [4,2,3,5,6,1] => [6,1,3,2,4,5] => 7

{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [6,4,5,2,1,3] => [5,1,6,2,4,3] => 11

{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [4,2,3,5,1,6] => [5,1,3,2,4,6] => 6

{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [4,5,6,2,3,1] => [6,1,5,2,3,4] => 9

{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [4,5,2,6,1,3] => [1,5,6,2,3,4] => 8

{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [4,5,2,1,3,6] => [4,1,5,2,3,6] => 6

{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [4,6,2,3,5,1] => [6,1,2,4,5,3] => 9

{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [6,4,2,5,1,3] => [1,5,6,2,4,3] => 11

{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [4,2,3,6,1,5] => [1,3,5,4,6,2] => 8

{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [4,2,3,1,5,6] => [4,1,3,2,5,6] => 5

{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [5,4,2,3,6,1] => [6,1,4,3,2,5] => 10

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + q + 2\ q^{2} + q^{3}$

$F_{4} = 1 + q + 2\ q^{2} + 5\ q^{3} + 2\ q^{4} + 3\ q^{5} + q^{6}$

$F_{5} = 1 + q + 2\ q^{2} + 5\ q^{3} + 8\ q^{4} + 4\ q^{5} + 10\ q^{6} + 12\ q^{7} + 4\ q^{8} + 4\ q^{9} + q^{10}$

$F_{6} = 1 + q + 2\ q^{2} + 5\ q^{3} + 8\ q^{4} + 12\ q^{5} + 12\ q^{6} + 23\ q^{7} + 26\ q^{8} + 26\ q^{9} + 22\ q^{10} + 29\ q^{11} + 23\ q^{12} + 7\ q^{13} + 5\ q^{14} + q^{15}$

Description

The Rajchgot index of a permutation.
The Rajchgot index of a permutation

$\sigma$ is the degree of the Grothendieck polynomial of

$\sigma$ . This statistic on permutations was defined by Pechenik, Speyer, and Weigandt [1]. It can be computed by taking the maximum major index St000004The major index of a permutation. of the permutations smaller than or equal to

$\sigma$ in the right weak Bruhat order.

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00067Foata bijection.

Map

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.