- FindStat

Identifier

Mp00109: Permutations —descent word⟶ Binary words
St000529: Binary words ⟶ ℤ

Values

[1,2] => 0 => 1

[2,1] => 1 => 1

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

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

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

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

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

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

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

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

[1,3,2,4] => 010 => 5

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

[1,4,2,3] => 010 => 5

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

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

[2,1,4,3] => 101 => 5

[2,3,1,4] => 010 => 5

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

[2,4,1,3] => 010 => 5

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

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

[3,1,4,2] => 101 => 5

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

[3,2,4,1] => 101 => 5

[3,4,1,2] => 010 => 5

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

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

[4,1,3,2] => 101 => 5

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

[4,2,3,1] => 101 => 5

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

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

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

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

[1,2,4,3,5] => 0010 => 9

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

[1,2,5,3,4] => 0010 => 9

[1,2,5,4,3] => 0011 => 6

[1,3,2,4,5] => 0100 => 9

[1,3,2,5,4] => 0101 => 16

[1,3,4,2,5] => 0010 => 9

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

[1,3,5,2,4] => 0010 => 9

[1,3,5,4,2] => 0011 => 6

[1,4,2,3,5] => 0100 => 9

[1,4,2,5,3] => 0101 => 16

[1,4,3,2,5] => 0110 => 11

[1,4,3,5,2] => 0101 => 16

[1,4,5,2,3] => 0010 => 9

[1,4,5,3,2] => 0011 => 6

[1,5,2,3,4] => 0100 => 9

[1,5,2,4,3] => 0101 => 16

[1,5,3,2,4] => 0110 => 11

[1,5,3,4,2] => 0101 => 16

[1,5,4,2,3] => 0110 => 11

[1,5,4,3,2] => 0111 => 4

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

[2,1,3,5,4] => 1001 => 11

[2,1,4,3,5] => 1010 => 16

[2,1,4,5,3] => 1001 => 11

[2,1,5,3,4] => 1010 => 16

[2,1,5,4,3] => 1011 => 9

[2,3,1,4,5] => 0100 => 9

[2,3,1,5,4] => 0101 => 16

[2,3,4,1,5] => 0010 => 9

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

[2,3,5,1,4] => 0010 => 9

[2,3,5,4,1] => 0011 => 6

[2,4,1,3,5] => 0100 => 9

[2,4,1,5,3] => 0101 => 16

[2,4,3,1,5] => 0110 => 11

[2,4,3,5,1] => 0101 => 16

[2,4,5,1,3] => 0010 => 9

[2,4,5,3,1] => 0011 => 6

[2,5,1,3,4] => 0100 => 9

[2,5,1,4,3] => 0101 => 16

[2,5,3,1,4] => 0110 => 11

[2,5,3,4,1] => 0101 => 16

[2,5,4,1,3] => 0110 => 11

[2,5,4,3,1] => 0111 => 4

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

[3,1,2,5,4] => 1001 => 11

[3,1,4,2,5] => 1010 => 16

[3,1,4,5,2] => 1001 => 11

[3,1,5,2,4] => 1010 => 16

[3,1,5,4,2] => 1011 => 9

[3,2,1,4,5] => 1100 => 6

[3,2,1,5,4] => 1101 => 9

[3,2,4,1,5] => 1010 => 16

[3,2,4,5,1] => 1001 => 11

[3,2,5,1,4] => 1010 => 16

[3,2,5,4,1] => 1011 => 9

[3,4,1,2,5] => 0100 => 9

[3,4,1,5,2] => 0101 => 16

[3,4,2,1,5] => 0110 => 11

[3,4,2,5,1] => 0101 => 16

[3,4,5,1,2] => 0010 => 9

[3,4,5,2,1] => 0011 => 6

[3,5,1,2,4] => 0100 => 9

[3,5,1,4,2] => 0101 => 16

[3,5,2,1,4] => 0110 => 11

>>> Load all 33030 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{2} = 2\ q$

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

$F_{4} = 2\ q + 12\ q^{3} + 10\ q^{5}$

$F_{5} = 2\ q + 16\ q^{4} + 12\ q^{6} + 36\ q^{9} + 22\ q^{11} + 32\ q^{16}$

$F_{6} = 2\ q + 20\ q^{5} + 40\ q^{10} + 56\ q^{14} + 76\ q^{19} + 104\ q^{26} + 140\ q^{35} + 160\ q^{40} + 122\ q^{61}$

$F_{7} = 2\ q + 24\ q^{6} + 60\ q^{15} + 120\ q^{20} + 58\ q^{29} + 136\ q^{34} + 200\ q^{50} + 220\ q^{55} + 256\ q^{64} + 142\ q^{71} + 312\ q^{78} + 180\ q^{90} + 396\ q^{99} + 444\ q^{111} + 264\ q^{132} + 620\ q^{155} + 338\ q^{169} + 724\ q^{181} + 544\ q^{272}$

Description

The number of permutations whose descent word is the given binary word.
This is the sizes of the preimages of the map Mp00109descent word.

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.