- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000004: 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] => [3,1,2] => [3,2,1] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => [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] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => 15

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

{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [6,1,2,3,5,4] => [5,6,4,3,2,1] => 14

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

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

{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [6,1,2,4,3,5] => [4,6,5,3,2,1] => 14

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

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

{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [6,1,2,5,4,3] => [5,4,6,3,2,1] => 13

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

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

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

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

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

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

{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [6,1,3,2,4,5] => [3,6,5,4,2,1] => 14

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

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

{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [6,1,5,2,3,4] => [5,3,6,4,2,1] => 13

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

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

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

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

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

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

{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [6,1,4,3,2,5] => [4,3,6,5,2,1] => 13

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

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

$F_{6} = 1 + q + 2\ q^{2} + 4\ q^{3} + 7\ q^{4} + 12\ q^{5} + 14\ q^{6} + 20\ q^{7} + 26\ q^{8} + 29\ q^{9} + 26\ q^{10} + 25\ q^{11} + 20\ q^{12} + 10\ q^{13} + 5\ q^{14} + q^{15}$

Description

The major index of a permutation.
This is the sum of the positions of its descents,

$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$
Its generating function is

$[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for

$[k]_q = 1 + q + q^2 + \dots q^{k-1}$ .
A statistic equidistributed with the major index is called Mahonian statistic.

Map

to permutation

Description

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

Map

inverse

Description

Sends a permutation to its inverse.

Map

inverse first fundamental transformation

Description

Let

$\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps

$\sigma$ to the permutation

$[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.