- FindStat

Identifier

Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤ

Values

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

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

[1,2,3] => [1,2,3] => 00 => 0

[1,3,2] => [3,1,2] => 10 => 1

[2,1,3] => [2,1,3] => 10 => 1

[2,3,1] => [2,3,1] => 01 => 2

[3,1,2] => [1,3,2] => 01 => 2

[3,2,1] => [3,2,1] => 11 => 3

[1,2,3,4] => [1,2,3,4] => 000 => 0

[1,2,4,3] => [4,1,2,3] => 100 => 1

[1,3,2,4] => [3,1,2,4] => 100 => 1

[1,3,4,2] => [3,4,1,2] => 010 => 2

[1,4,2,3] => [1,4,2,3] => 010 => 2

[1,4,3,2] => [4,3,1,2] => 110 => 3

[2,1,3,4] => [2,1,3,4] => 100 => 1

[2,1,4,3] => [2,4,1,3] => 010 => 2

[2,3,1,4] => [2,3,1,4] => 010 => 2

[2,3,4,1] => [2,3,4,1] => 001 => 3

[2,4,1,3] => [4,2,1,3] => 110 => 3

[2,4,3,1] => [4,2,3,1] => 101 => 4

[3,1,2,4] => [1,3,2,4] => 010 => 2

[3,1,4,2] => [1,3,4,2] => 001 => 3

[3,2,1,4] => [3,2,1,4] => 110 => 3

[3,2,4,1] => [3,2,4,1] => 101 => 4

[3,4,1,2] => [3,1,4,2] => 101 => 4

[3,4,2,1] => [3,4,2,1] => 011 => 5

[4,1,2,3] => [1,2,4,3] => 001 => 3

[4,1,3,2] => [4,1,3,2] => 101 => 4

[4,2,1,3] => [2,1,4,3] => 101 => 4

[4,2,3,1] => [2,4,3,1] => 011 => 5

[4,3,1,2] => [1,4,3,2] => 011 => 5

[4,3,2,1] => [4,3,2,1] => 111 => 6

[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0

[1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1

[1,2,4,3,5] => [4,1,2,3,5] => 1000 => 1

[1,2,4,5,3] => [4,5,1,2,3] => 0100 => 2

[1,2,5,3,4] => [1,5,2,3,4] => 0100 => 2

[1,2,5,4,3] => [5,4,1,2,3] => 1100 => 3

[1,3,2,4,5] => [3,1,2,4,5] => 1000 => 1

[1,3,2,5,4] => [3,5,1,2,4] => 0100 => 2

[1,3,4,2,5] => [3,4,1,2,5] => 0100 => 2

[1,3,4,5,2] => [3,4,5,1,2] => 0010 => 3

[1,3,5,2,4] => [5,3,1,2,4] => 1100 => 3

[1,3,5,4,2] => [5,3,4,1,2] => 1010 => 4

[1,4,2,3,5] => [1,4,2,3,5] => 0100 => 2

[1,4,2,5,3] => [1,4,5,2,3] => 0010 => 3

[1,4,3,2,5] => [4,3,1,2,5] => 1100 => 3

[1,4,3,5,2] => [4,3,5,1,2] => 1010 => 4

[1,4,5,2,3] => [4,1,5,2,3] => 1010 => 4

[1,4,5,3,2] => [4,5,3,1,2] => 0110 => 5

[1,5,2,3,4] => [1,2,5,3,4] => 0010 => 3

[1,5,2,4,3] => [5,1,4,2,3] => 1010 => 4

[1,5,3,2,4] => [5,1,3,2,4] => 1010 => 4

[1,5,3,4,2] => [3,5,4,1,2] => 0110 => 5

[1,5,4,2,3] => [1,5,4,2,3] => 0110 => 5

[1,5,4,3,2] => [5,4,3,1,2] => 1110 => 6

[2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1

[2,1,3,5,4] => [2,5,1,3,4] => 0100 => 2

[2,1,4,3,5] => [2,4,1,3,5] => 0100 => 2

[2,1,4,5,3] => [2,4,5,1,3] => 0010 => 3

[2,1,5,3,4] => [5,2,1,3,4] => 1100 => 3

[2,1,5,4,3] => [5,2,4,1,3] => 1010 => 4

[2,3,1,4,5] => [2,3,1,4,5] => 0100 => 2

[2,3,1,5,4] => [2,3,5,1,4] => 0010 => 3

[2,3,4,1,5] => [2,3,4,1,5] => 0010 => 3

[2,3,4,5,1] => [2,3,4,5,1] => 0001 => 4

[2,3,5,1,4] => [5,2,3,1,4] => 1010 => 4

[2,3,5,4,1] => [5,2,3,4,1] => 1001 => 5

[2,4,1,3,5] => [4,2,1,3,5] => 1100 => 3

[2,4,1,5,3] => [4,2,5,1,3] => 1010 => 4

[2,4,3,1,5] => [4,2,3,1,5] => 1010 => 4

[2,4,3,5,1] => [4,2,3,5,1] => 1001 => 5

[2,4,5,1,3] => [4,5,2,1,3] => 0110 => 5

[2,4,5,3,1] => [4,5,2,3,1] => 0101 => 6

[2,5,1,3,4] => [2,1,5,3,4] => 1010 => 4

[2,5,1,4,3] => [2,5,4,1,3] => 0110 => 5

[2,5,3,1,4] => [2,5,3,1,4] => 0110 => 5

[2,5,3,4,1] => [2,5,3,4,1] => 0101 => 6

[2,5,4,1,3] => [5,4,2,1,3] => 1110 => 6

[2,5,4,3,1] => [5,4,2,3,1] => 1101 => 7

[3,1,2,4,5] => [1,3,2,4,5] => 0100 => 2

[3,1,2,5,4] => [1,3,5,2,4] => 0010 => 3

[3,1,4,2,5] => [1,3,4,2,5] => 0010 => 3

[3,1,4,5,2] => [1,3,4,5,2] => 0001 => 4

[3,1,5,2,4] => [3,1,5,2,4] => 1010 => 4

[3,1,5,4,2] => [5,1,3,4,2] => 1001 => 5

[3,2,1,4,5] => [3,2,1,4,5] => 1100 => 3

[3,2,1,5,4] => [3,2,5,1,4] => 1010 => 4

[3,2,4,1,5] => [3,2,4,1,5] => 1010 => 4

[3,2,4,5,1] => [3,2,4,5,1] => 1001 => 5

[3,2,5,1,4] => [3,5,2,1,4] => 0110 => 5

[3,2,5,4,1] => [3,5,2,4,1] => 0101 => 6

[3,4,1,2,5] => [3,1,4,2,5] => 1010 => 4

[3,4,1,5,2] => [3,1,4,5,2] => 1001 => 5

[3,4,2,1,5] => [3,4,2,1,5] => 0110 => 5

[3,4,2,5,1] => [3,4,2,5,1] => 0101 => 6

[3,4,5,1,2] => [3,4,1,5,2] => 0101 => 6

[3,4,5,2,1] => [3,4,5,2,1] => 0011 => 7

[3,5,1,2,4] => [1,5,3,2,4] => 0110 => 5

[3,5,1,4,2] => [3,5,1,4,2] => 0101 => 6

[3,5,2,1,4] => [5,3,2,1,4] => 1110 => 6

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$F_{2} = 1 + q$

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

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

$F_{5} = 1 + 4\ q + 9\ q^{2} + 15\ q^{3} + 20\ q^{4} + 22\ q^{5} + 20\ q^{6} + 15\ q^{7} + 9\ q^{8} + 4\ q^{9} + q^{10}$

$F_{6} = 1 + 5\ q + 14\ q^{2} + 29\ q^{3} + 49\ q^{4} + 71\ q^{5} + 90\ q^{6} + 101\ q^{7} + 101\ q^{8} + 90\ q^{9} + 71\ q^{10} + 49\ q^{11} + 29\ q^{12} + 14\ q^{13} + 5\ q^{14} + q^{15}$

$F_{7} = 1 + 6\ q + 20\ q^{2} + 49\ q^{3} + 98\ q^{4} + 169\ q^{5} + 259\ q^{6} + 359\ q^{7} + 455\ q^{8} + 531\ q^{9} + 573\ q^{10} + 573\ q^{11} + 531\ q^{12} + 455\ q^{13} + 359\ q^{14} + 259\ q^{15} + 169\ q^{16} + 98\ q^{17} + 49\ q^{18} + 20\ q^{19} + 6\ q^{20} + q^{21}$

Description

The sum of the positions of the ones in a binary word.

Map

inverse Foata bijection

Description

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

Map

descent word

Description

The descent positions of a permutation as a binary word.
For a permutation

$\pi$ of

$n$ letters and each

$1\leq i\leq n-1$ such that

$\pi(i) > \pi(i+1)$ we set

$w_i=1$ , otherwise

$w_i=0$ .
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.