- FindStat

Identifier

Mp00131: Permutations —descent bottoms⟶ Binary words
St000289: Binary words ⟶ ℤ

Values

[1,2] => 0 => 0

[2,1] => 1 => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,1,4,2] => 110 => 6

[3,2,1,4] => 110 => 6

[3,2,4,1] => 110 => 6

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

[3,4,2,1] => 110 => 6

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

[4,1,3,2] => 110 => 6

[4,2,1,3] => 110 => 6

[4,2,3,1] => 110 => 6

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

[4,3,2,1] => 111 => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,4,2,5,3] => 0110 => 6

[1,4,3,2,5] => 0110 => 6

[1,4,3,5,2] => 0110 => 6

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

[1,4,5,3,2] => 0110 => 6

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

[1,5,2,4,3] => 0110 => 6

[1,5,3,2,4] => 0110 => 6

[1,5,3,4,2] => 0110 => 6

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

[1,5,4,3,2] => 0111 => 7

[2,1,3,4,5] => 1000 => 8

[2,1,3,5,4] => 1001 => 9

[2,1,4,3,5] => 1010 => 10

[2,1,4,5,3] => 1010 => 10

[2,1,5,3,4] => 1010 => 10

[2,1,5,4,3] => 1011 => 11

[2,3,1,4,5] => 1000 => 8

[2,3,1,5,4] => 1001 => 9

[2,3,4,1,5] => 1000 => 8

[2,3,4,5,1] => 1000 => 8

[2,3,5,1,4] => 1000 => 8

[2,3,5,4,1] => 1001 => 9

[2,4,1,3,5] => 1000 => 8

[2,4,1,5,3] => 1010 => 10

[2,4,3,1,5] => 1010 => 10

[2,4,3,5,1] => 1010 => 10

[2,4,5,1,3] => 1000 => 8

[2,4,5,3,1] => 1010 => 10

[2,5,1,3,4] => 1000 => 8

[2,5,1,4,3] => 1010 => 10

[2,5,3,1,4] => 1010 => 10

[2,5,3,4,1] => 1010 => 10

[2,5,4,1,3] => 1001 => 9

[2,5,4,3,1] => 1011 => 11

[3,1,2,4,5] => 1000 => 8

[3,1,2,5,4] => 1001 => 9

[3,1,4,2,5] => 1100 => 12

[3,1,4,5,2] => 1100 => 12

[3,1,5,2,4] => 1100 => 12

[3,1,5,4,2] => 1101 => 13

[3,2,1,4,5] => 1100 => 12

[3,2,1,5,4] => 1101 => 13

[3,2,4,1,5] => 1100 => 12

[3,2,4,5,1] => 1100 => 12

[3,2,5,1,4] => 1100 => 12

[3,2,5,4,1] => 1101 => 13

[3,4,1,2,5] => 1000 => 8

[3,4,1,5,2] => 1100 => 12

[3,4,2,1,5] => 1100 => 12

[3,4,2,5,1] => 1100 => 12

[3,4,5,1,2] => 1000 => 8

[3,4,5,2,1] => 1100 => 12

[3,5,1,2,4] => 1000 => 8

[3,5,1,4,2] => 1100 => 12

[3,5,2,1,4] => 1100 => 12

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$F_{2} = 1 + q$

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

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

$F_{5} = 1 + q + 3\ q^{2} + q^{3} + 7\ q^{4} + 3\ q^{5} + 7\ q^{6} + q^{7} + 15\ q^{8} + 7\ q^{9} + 17\ q^{10} + 3\ q^{11} + 31\ q^{12} + 7\ q^{13} + 15\ q^{14} + q^{15}$

$F_{6} = 1 + q + 3\ q^{2} + q^{3} + 7\ q^{4} + 3\ q^{5} + 7\ q^{6} + q^{7} + 15\ q^{8} + 7\ q^{9} + 17\ q^{10} + 3\ q^{11} + 31\ q^{12} + 7\ q^{13} + 15\ q^{14} + q^{15} + 31\ q^{16} + 15\ q^{17} + 37\ q^{18} + 7\ q^{19} + 69\ q^{20} + 17\ q^{21} + 37\ q^{22} + 3\ q^{23} + 115\ q^{24} + 31\ q^{25} + 69\ q^{26} + 7\ q^{27} + 115\ q^{28} + 15\ q^{29} + 31\ q^{30} + q^{31}$

$F_{7} = 1 + q + 3\ q^{2} + q^{3} + 7\ q^{4} + 3\ q^{5} + 7\ q^{6} + q^{7} + 15\ q^{8} + 7\ q^{9} + 17\ q^{10} + 3\ q^{11} + 31\ q^{12} + 7\ q^{13} + 15\ q^{14} + q^{15} + 31\ q^{16} + 15\ q^{17} + 37\ q^{18} + 7\ q^{19} + 69\ q^{20} + 17\ q^{21} + 37\ q^{22} + 3\ q^{23} + 115\ q^{24} + 31\ q^{25} + 69\ q^{26} + 7\ q^{27} + 115\ q^{28} + 15\ q^{29} + 31\ q^{30} + q^{31} + 63\ q^{32} + 31\ q^{33} + 77\ q^{34} + 15\ q^{35} + 145\ q^{36} + 37\ q^{37} + 81\ q^{38} + 7\ q^{39} + 245\ q^{40} + 69\ q^{41} + 155\ q^{42} + 17\ q^{43} + 261\ q^{44} + 37\ q^{45} + 77\ q^{46} + 3\ q^{47} + 391\ q^{48} + 115\ q^{49} + 261\ q^{50} + 31\ q^{51} + 445\ q^{52} + 69\ q^{53} + 145\ q^{54} + 7\ q^{55} + 675\ q^{56} + 115\ q^{57} + 245\ q^{58} + 15\ q^{59} + 391\ q^{60} + 31\ q^{61} + 63\ q^{62} + q^{63}$

Description

The decimal representation of a binary word.

Map

descent bottoms

Description

The descent bottoms of a permutation as a binary word.