- FindStat

Identifier

Mp00252: Permutations —restriction⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001838: Binary words ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

[1,3,2,4] => [1,3,2] => [2,1] => 1010 => 6

[1,3,4,2] => [1,3,2] => [2,1] => 1010 => 6

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

[1,4,3,2] => [1,3,2] => [2,1] => 1010 => 6

[2,1,3,4] => [2,1,3] => [2,1] => 1010 => 6

[2,1,4,3] => [2,1,3] => [2,1] => 1010 => 6

[2,3,1,4] => [2,3,1] => [2,1] => 1010 => 6

[2,3,4,1] => [2,3,1] => [2,1] => 1010 => 6

[2,4,1,3] => [2,1,3] => [2,1] => 1010 => 6

[2,4,3,1] => [2,3,1] => [2,1] => 1010 => 6

[3,1,2,4] => [3,1,2] => [2,1] => 1010 => 6

[3,1,4,2] => [3,1,2] => [2,1] => 1010 => 6

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

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

[3,4,1,2] => [3,1,2] => [2,1] => 1010 => 6

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

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

[4,1,3,2] => [1,3,2] => [2,1] => 1010 => 6

[4,2,1,3] => [2,1,3] => [2,1] => 1010 => 6

[4,2,3,1] => [2,3,1] => [2,1] => 1010 => 6

[4,3,1,2] => [3,1,2] => [2,1] => 1010 => 6

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

[1,2,3,4,5] => [1,2,3,4] => [4] => 10000 => 6

[1,2,3,5,4] => [1,2,3,4] => [4] => 10000 => 6

[1,2,4,3,5] => [1,2,4,3] => [3,1] => 10010 => 10

[1,2,4,5,3] => [1,2,4,3] => [3,1] => 10010 => 10

[1,2,5,3,4] => [1,2,3,4] => [4] => 10000 => 6

[1,2,5,4,3] => [1,2,4,3] => [3,1] => 10010 => 10

[1,3,2,4,5] => [1,3,2,4] => [3,1] => 10010 => 10

[1,3,2,5,4] => [1,3,2,4] => [3,1] => 10010 => 10

[1,3,4,2,5] => [1,3,4,2] => [3,1] => 10010 => 10

[1,3,4,5,2] => [1,3,4,2] => [3,1] => 10010 => 10

[1,3,5,2,4] => [1,3,2,4] => [3,1] => 10010 => 10

[1,3,5,4,2] => [1,3,4,2] => [3,1] => 10010 => 10

[1,4,2,3,5] => [1,4,2,3] => [3,1] => 10010 => 10

[1,4,2,5,3] => [1,4,2,3] => [3,1] => 10010 => 10

[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => 10110 => 10

[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => 10110 => 10

[1,4,5,2,3] => [1,4,2,3] => [3,1] => 10010 => 10

[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => 10110 => 10

[1,5,2,3,4] => [1,2,3,4] => [4] => 10000 => 6

[1,5,2,4,3] => [1,2,4,3] => [3,1] => 10010 => 10

[1,5,3,2,4] => [1,3,2,4] => [3,1] => 10010 => 10

[1,5,3,4,2] => [1,3,4,2] => [3,1] => 10010 => 10

[1,5,4,2,3] => [1,4,2,3] => [3,1] => 10010 => 10

[1,5,4,3,2] => [1,4,3,2] => [2,1,1] => 10110 => 10

[2,1,3,4,5] => [2,1,3,4] => [3,1] => 10010 => 10

[2,1,3,5,4] => [2,1,3,4] => [3,1] => 10010 => 10

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

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

[2,1,5,3,4] => [2,1,3,4] => [3,1] => 10010 => 10

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

[2,3,1,4,5] => [2,3,1,4] => [3,1] => 10010 => 10

[2,3,1,5,4] => [2,3,1,4] => [3,1] => 10010 => 10

[2,3,4,1,5] => [2,3,4,1] => [3,1] => 10010 => 10

[2,3,4,5,1] => [2,3,4,1] => [3,1] => 10010 => 10

[2,3,5,1,4] => [2,3,1,4] => [3,1] => 10010 => 10

[2,3,5,4,1] => [2,3,4,1] => [3,1] => 10010 => 10

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

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

[2,4,3,1,5] => [2,4,3,1] => [2,1,1] => 10110 => 10

[2,4,3,5,1] => [2,4,3,1] => [2,1,1] => 10110 => 10

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

[2,4,5,3,1] => [2,4,3,1] => [2,1,1] => 10110 => 10

[2,5,1,3,4] => [2,1,3,4] => [3,1] => 10010 => 10

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

[2,5,3,1,4] => [2,3,1,4] => [3,1] => 10010 => 10

[2,5,3,4,1] => [2,3,4,1] => [3,1] => 10010 => 10

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

[2,5,4,3,1] => [2,4,3,1] => [2,1,1] => 10110 => 10

[3,1,2,4,5] => [3,1,2,4] => [3,1] => 10010 => 10

[3,1,2,5,4] => [3,1,2,4] => [3,1] => 10010 => 10

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

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

[3,1,5,2,4] => [3,1,2,4] => [3,1] => 10010 => 10

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

[3,2,1,4,5] => [3,2,1,4] => [2,1,1] => 10110 => 10

[3,2,1,5,4] => [3,2,1,4] => [2,1,1] => 10110 => 10

[3,2,4,1,5] => [3,2,4,1] => [2,1,1] => 10110 => 10

[3,2,4,5,1] => [3,2,4,1] => [2,1,1] => 10110 => 10

[3,2,5,1,4] => [3,2,1,4] => [2,1,1] => 10110 => 10

[3,2,5,4,1] => [3,2,4,1] => [2,1,1] => 10110 => 10

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

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

[3,4,2,1,5] => [3,4,2,1] => [2,1,1] => 10110 => 10

[3,4,2,5,1] => [3,4,2,1] => [2,1,1] => 10110 => 10

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

[3,4,5,2,1] => [3,4,2,1] => [2,1,1] => 10110 => 10

[3,5,1,2,4] => [3,1,2,4] => [3,1] => 10010 => 10

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

[3,5,2,1,4] => [3,2,1,4] => [2,1,1] => 10110 => 10

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

$F_{3} = 6\ q^{4}$

$F_{4} = 8\ q^{5} + 16\ q^{6}$

$F_{5} = 30\ q^{6} + 90\ q^{10}$

$F_{6} = 12\ q^{7} + 300\ q^{9} + 192\ q^{13} + 216\ q^{14}$

$F_{7} = 2156\ q^{8} + 1134\ q^{12} + 350\ q^{16} + 1400\ q^{18}$

Description

The number of nonempty primitive factors of a binary word.
A word

$u$ is a factor of a word

$w$ if

$w = p u s$ for words

$p$ and

$s$ . A word is primitive, if it is not of the form

$u^k$ for a word

$u$ and an integer

$k\geq 2$ .
Apparently, the maximal number of nonempty primitive factors a binary word of length

$n$ can have is given by oeis:A131673.

Map

Robinson-Schensted tableau shape

Description

Sends a permutation to its Robinson-Schensted tableau shape.
The Robinson-Schensted corrspondence is a bijection between permutations of length

$n$ and pairs of standard Young tableaux of the same shape and of size

$n$ , see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to the shape of its corresponding insertion and recording tableau.

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.

Map

to binary word

Description

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.