- FindStat

Identifier

Mp00131: Permutations —descent bottoms⟶ Binary words
St000391: Binary words ⟶ ℤ

Values

[1,2] => 0 => 0

[2,1] => 1 => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,5,4,3,2] => 0111 => 9

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

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

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

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

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

[2,1,5,4,3] => 1011 => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,5,4,3,1] => 1011 => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 1 + q$

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

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

$F_{5} = 1 + 15\ q + 7\ q^{2} + 34\ q^{3} + 18\ q^{4} + 14\ q^{5} + 18\ q^{6} + 8\ q^{7} + 3\ q^{8} + q^{9} + q^{10}$

$F_{6} = 1 + 31\ q + 15\ q^{2} + 122\ q^{3} + 72\ q^{4} + 69\ q^{5} + 147\ q^{6} + 83\ q^{7} + 71\ q^{8} + 33\ q^{9} + 45\ q^{10} + 18\ q^{11} + 8\ q^{12} + 3\ q^{13} + q^{14} + q^{15}$

$F_{7} = 1 + 63\ q + 31\ q^{2} + 406\ q^{3} + 252\ q^{4} + 263\ q^{5} + 822\ q^{6} + 544\ q^{7} + 554\ q^{8} + 399\ q^{9} + 613\ q^{10} + 351\ q^{11} + 307\ q^{12} + 160\ q^{13} + 102\ q^{14} + 96\ q^{15} + 45\ q^{16} + 18\ q^{17} + 8\ q^{18} + 3\ q^{19} + q^{20} + q^{21}$

Description

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

Map

descent bottoms

Description

The descent bottoms of a permutation as a binary word.