- FindStat

Your data matches 14 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000510
(load all 2 compositions to match this statistic)

Mapping

St000510: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2]
 => 1
[1,1]
 => 1
[3]
 => 2
[2,1]
 => 0
[1,1,1]
 => 2
[4]
 => 2
[3,1]
 => 0
[2,2]
 => 2
[2,1,1]
 => 0
[1,1,1,1]
 => 6
[5]
 => 4
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 0
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 24
[6]
 => 2
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 0
[3,3]
 => 6
[3,2,1]
 => 0
[3,1,1,1]
 => 0
[2,2,2]
 => 8
[2,2,1,1]
 => 0
[2,1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => 120
[7]
 => 6
[6,1]
 => 0
[5,2]
 => 0
[5,1,1]
 => 0
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 0
[3,3,1]
 => 0
[3,2,2]
 => 0
[3,2,1,1]
 => 0
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 720
[8]
 => 4
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 0
[5,3]
 => 0
[5,2,1]
 => 0
[5,1,1,1]
 => 0

Description

The number of invariant oriented cycles when acting with a permutation of given cycle type.

Matching statistic: St000296

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 83%●distinct values known / distinct values provided: 4%

Values

[2]
 => []
 => ?
 => ? => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 =>  => ? ∊ {1,1}
[3]
 => []
 => ?
 => ? => ? ∊ {2,2}
[2,1]
 => [1]
 => []
 =>  => ? ∊ {2,2}
[1,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[4]
 => []
 => ?
 => ? => ? ∊ {2,2,6}
[3,1]
 => [1]
 => []
 =>  => ? ∊ {2,2,6}
[2,2]
 => [2]
 => []
 =>  => ? ∊ {2,2,6}
[2,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[5]
 => []
 => ?
 => ? => ? ∊ {0,4,24}
[4,1]
 => [1]
 => []
 =>  => ? ∊ {0,4,24}
[3,2]
 => [2]
 => []
 =>  => ? ∊ {0,4,24}
[3,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[2,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[6]
 => []
 => ?
 => ? => ? ∊ {2,6,8,120}
[5,1]
 => [1]
 => []
 =>  => ? ∊ {2,6,8,120}
[4,2]
 => [2]
 => []
 =>  => ? ∊ {2,6,8,120}
[4,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[3,3]
 => [3]
 => []
 =>  => ? ∊ {2,6,8,120}
[3,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[2,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[7]
 => []
 => ?
 => ? => ? ∊ {0,0,6,720}
[6,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,6,720}
[5,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,6,720}
[5,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[4,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,6,720}
[4,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[3,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[3,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[8]
 => []
 => ?
 => ? => ? ∊ {0,4,8,48,5040}
[7,1]
 => [1]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[6,2]
 => [2]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[6,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[5,3]
 => [3]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[5,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[4,4]
 => [4]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[4,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[4,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[3,3,2]
 => [3,2]
 => [2]
 => 100 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => 110 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => 1100 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => 10110 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 0
[9]
 => []
 => ?
 => ? => ? ∊ {0,0,6,36,40320}
[8,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[7,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[7,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[6,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[6,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[5,4]
 => [4]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[5,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[5,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[4,4,1]
 => [4,1]
 => [1]
 => 10 => 0
[10]
 => []
 => ?
 => ? => ? ∊ {0,0,4,20,384,362880}
[9,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[8,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[7,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[6,4]
 => [4]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[5,5]
 => [5]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[11]
 => []
 => ?
 => ? => ? ∊ {0,0,0,0,10,3628800}
[10,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,0,0,10,3628800}
[9,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,0,0,10,3628800}
[8,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,0,0,10,3628800}
[7,4]
 => [4]
 => []
 =>  => ? ∊ {0,0,0,0,10,3628800}
[6,5]
 => [5]
 => []
 =>  => ? ∊ {0,0,0,0,10,3628800}
[12]
 => []
 => ?
 => ? => ? ∊ {0,4,12,64,324,3840,39916800}
[11,1]
 => [1]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}
[10,2]
 => [2]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}
[9,3]
 => [3]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}
[8,4]
 => [4]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}
[7,5]
 => [5]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}
[6,6]
 => [6]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}

Description

The length of the symmetric border of a binary word. The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix. The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].

Matching statistic: St000629
(load all 2 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 83%●distinct values known / distinct values provided: 4%

Values

[2]
 => []
 => ?
 => ? => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 =>  => ? ∊ {1,1}
[3]
 => []
 => ?
 => ? => ? ∊ {2,2}
[2,1]
 => [1]
 => []
 =>  => ? ∊ {2,2}
[1,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[4]
 => []
 => ?
 => ? => ? ∊ {2,2,6}
[3,1]
 => [1]
 => []
 =>  => ? ∊ {2,2,6}
[2,2]
 => [2]
 => []
 =>  => ? ∊ {2,2,6}
[2,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[5]
 => []
 => ?
 => ? => ? ∊ {0,4,24}
[4,1]
 => [1]
 => []
 =>  => ? ∊ {0,4,24}
[3,2]
 => [2]
 => []
 =>  => ? ∊ {0,4,24}
[3,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[2,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[6]
 => []
 => ?
 => ? => ? ∊ {2,6,8,120}
[5,1]
 => [1]
 => []
 =>  => ? ∊ {2,6,8,120}
[4,2]
 => [2]
 => []
 =>  => ? ∊ {2,6,8,120}
[4,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[3,3]
 => [3]
 => []
 =>  => ? ∊ {2,6,8,120}
[3,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[2,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[7]
 => []
 => ?
 => ? => ? ∊ {0,0,6,720}
[6,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,6,720}
[5,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,6,720}
[5,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[4,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,6,720}
[4,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[3,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[3,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[8]
 => []
 => ?
 => ? => ? ∊ {0,4,8,48,5040}
[7,1]
 => [1]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[6,2]
 => [2]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[6,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[5,3]
 => [3]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[5,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[4,4]
 => [4]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[4,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[4,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[3,3,2]
 => [3,2]
 => [2]
 => 100 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => 110 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => 1100 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => 10110 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 0
[9]
 => []
 => ?
 => ? => ? ∊ {0,0,6,36,40320}
[8,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[7,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[7,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[6,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[6,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[5,4]
 => [4]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[5,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[5,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[4,4,1]
 => [4,1]
 => [1]
 => 10 => 0
[10]
 => []
 => ?
 => ? => ? ∊ {0,0,4,20,384,362880}
[9,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[8,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[7,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[6,4]
 => [4]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[5,5]
 => [5]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[11]
 => []
 => ?
 => ? => ? ∊ {0,0,0,0,10,3628800}
[10,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,0,0,10,3628800}
[9,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,0,0,10,3628800}
[8,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,0,0,10,3628800}
[7,4]
 => [4]
 => []
 =>  => ? ∊ {0,0,0,0,10,3628800}
[6,5]
 => [5]
 => []
 =>  => ? ∊ {0,0,0,0,10,3628800}
[12]
 => []
 => ?
 => ? => ? ∊ {0,4,12,64,324,3840,39916800}
[11,1]
 => [1]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}
[10,2]
 => [2]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}
[9,3]
 => [3]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}
[8,4]
 => [4]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}
[7,5]
 => [5]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}
[6,6]
 => [6]
 => []
 =>  => ? ∊ {0,4,12,64,324,3840,39916800}

Description

The defect of a binary word. The defect of a finite word

$w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in

$w$ . The maximum possible number of palindromic factors in a word

$w$ is

$|w|+1$ .

Matching statistic: St000921

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000921: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 82%●distinct values known / distinct values provided: 4%

Values

[2]
 => []
 => ?
 => ? => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 => ? => ? ∊ {1,1}
[3]
 => []
 => ?
 => ? => ? ∊ {2,2}
[2,1]
 => [1]
 => []
 => ? => ? ∊ {2,2}
[1,1,1]
 => [1,1]
 => [1]
 => 1 => 0
[4]
 => []
 => ?
 => ? => ? ∊ {2,2,6}
[3,1]
 => [1]
 => []
 => ? => ? ∊ {2,2,6}
[2,2]
 => [2]
 => []
 => ? => ? ∊ {2,2,6}
[2,1,1]
 => [1,1]
 => [1]
 => 1 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 11 => 0
[5]
 => []
 => ?
 => ? => ? ∊ {0,4,24}
[4,1]
 => [1]
 => []
 => ? => ? ∊ {0,4,24}
[3,2]
 => [2]
 => []
 => ? => ? ∊ {0,4,24}
[3,1,1]
 => [1,1]
 => [1]
 => 1 => 0
[2,2,1]
 => [2,1]
 => [1]
 => 1 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 11 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 111 => 0
[6]
 => []
 => ?
 => ? => ? ∊ {2,6,8,120}
[5,1]
 => [1]
 => []
 => ? => ? ∊ {2,6,8,120}
[4,2]
 => [2]
 => []
 => ? => ? ∊ {2,6,8,120}
[4,1,1]
 => [1,1]
 => [1]
 => 1 => 0
[3,3]
 => [3]
 => []
 => ? => ? ∊ {2,6,8,120}
[3,2,1]
 => [2,1]
 => [1]
 => 1 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 11 => 0
[2,2,2]
 => [2,2]
 => [2]
 => 0 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 11 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 111 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1111 => 0
[7]
 => []
 => ?
 => ? => ? ∊ {0,0,6,720}
[6,1]
 => [1]
 => []
 => ? => ? ∊ {0,0,6,720}
[5,2]
 => [2]
 => []
 => ? => ? ∊ {0,0,6,720}
[5,1,1]
 => [1,1]
 => [1]
 => 1 => 0
[4,3]
 => [3]
 => []
 => ? => ? ∊ {0,0,6,720}
[4,2,1]
 => [2,1]
 => [1]
 => 1 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 11 => 0
[3,3,1]
 => [3,1]
 => [1]
 => 1 => 0
[3,2,2]
 => [2,2]
 => [2]
 => 0 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 11 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 111 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 01 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 111 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1111 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 11111 => 0
[8]
 => []
 => ?
 => ? => ? ∊ {0,4,8,48,5040}
[7,1]
 => [1]
 => []
 => ? => ? ∊ {0,4,8,48,5040}
[6,2]
 => [2]
 => []
 => ? => ? ∊ {0,4,8,48,5040}
[6,1,1]
 => [1,1]
 => [1]
 => 1 => 0
[5,3]
 => [3]
 => []
 => ? => ? ∊ {0,4,8,48,5040}
[5,2,1]
 => [2,1]
 => [1]
 => 1 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => 11 => 0
[4,4]
 => [4]
 => []
 => ? => ? ∊ {0,4,8,48,5040}
[4,3,1]
 => [3,1]
 => [1]
 => 1 => 0
[4,2,2]
 => [2,2]
 => [2]
 => 0 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => 11 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 111 => 0
[3,3,2]
 => [3,2]
 => [2]
 => 0 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => 11 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => 01 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 111 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1111 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => 00 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => 011 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1111 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 11111 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 111111 => 0
[9]
 => []
 => ?
 => ? => ? ∊ {0,0,6,36,40320}
[8,1]
 => [1]
 => []
 => ? => ? ∊ {0,0,6,36,40320}
[7,2]
 => [2]
 => []
 => ? => ? ∊ {0,0,6,36,40320}
[7,1,1]
 => [1,1]
 => [1]
 => 1 => 0
[6,3]
 => [3]
 => []
 => ? => ? ∊ {0,0,6,36,40320}
[6,2,1]
 => [2,1]
 => [1]
 => 1 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => 11 => 0
[5,4]
 => [4]
 => []
 => ? => ? ∊ {0,0,6,36,40320}
[5,3,1]
 => [3,1]
 => [1]
 => 1 => 0
[5,2,2]
 => [2,2]
 => [2]
 => 0 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => 11 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 111 => 0
[4,4,1]
 => [4,1]
 => [1]
 => 1 => 0
[10]
 => []
 => ?
 => ? => ? ∊ {0,0,4,20,384,362880}
[9,1]
 => [1]
 => []
 => ? => ? ∊ {0,0,4,20,384,362880}
[8,2]
 => [2]
 => []
 => ? => ? ∊ {0,0,4,20,384,362880}
[7,3]
 => [3]
 => []
 => ? => ? ∊ {0,0,4,20,384,362880}
[6,4]
 => [4]
 => []
 => ? => ? ∊ {0,0,4,20,384,362880}
[5,5]
 => [5]
 => []
 => ? => ? ∊ {0,0,4,20,384,362880}
[11]
 => []
 => ?
 => ? => ? ∊ {0,0,0,0,10,3628800}
[10,1]
 => [1]
 => []
 => ? => ? ∊ {0,0,0,0,10,3628800}
[9,2]
 => [2]
 => []
 => ? => ? ∊ {0,0,0,0,10,3628800}
[8,3]
 => [3]
 => []
 => ? => ? ∊ {0,0,0,0,10,3628800}
[7,4]
 => [4]
 => []
 => ? => ? ∊ {0,0,0,0,10,3628800}
[6,5]
 => [5]
 => []
 => ? => ? ∊ {0,0,0,0,10,3628800}
[12]
 => []
 => ?
 => ? => ? ∊ {0,0,4,12,64,324,3840,39916800}
[11,1]
 => [1]
 => []
 => ? => ? ∊ {0,0,4,12,64,324,3840,39916800}
[10,2]
 => [2]
 => []
 => ? => ? ∊ {0,0,4,12,64,324,3840,39916800}
[9,3]
 => [3]
 => []
 => ? => ? ∊ {0,0,4,12,64,324,3840,39916800}
[8,4]
 => [4]
 => []
 => ? => ? ∊ {0,0,4,12,64,324,3840,39916800}
[7,5]
 => [5]
 => []
 => ? => ? ∊ {0,0,4,12,64,324,3840,39916800}
[6,6]
 => [6]
 => []
 => ? => ? ∊ {0,0,4,12,64,324,3840,39916800}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => 1111111111 => ? ∊ {0,0,4,12,64,324,3840,39916800}

Description

The number of internal inversions of a binary word. Let

$\bar w$ be the non-decreasing rearrangement of

$w$ , that is,

$\bar w$ is sorted. An internal inversion is a pair

$i < j$ such that

$w_i > w_j$ and

$\bar w_i = \bar w_j$ . For example, the word

$110$ has two inversions, but only the second is internal.

Matching statistic: St001371

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 81%●distinct values known / distinct values provided: 4%

Values

[2]
 => []
 => ?
 => ? => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 =>  => ? ∊ {1,1}
[3]
 => []
 => ?
 => ? => ? ∊ {2,2}
[2,1]
 => [1]
 => []
 =>  => ? ∊ {2,2}
[1,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[4]
 => []
 => ?
 => ? => ? ∊ {2,2,6}
[3,1]
 => [1]
 => []
 =>  => ? ∊ {2,2,6}
[2,2]
 => [2]
 => []
 =>  => ? ∊ {2,2,6}
[2,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[5]
 => []
 => ?
 => ? => ? ∊ {0,4,24}
[4,1]
 => [1]
 => []
 =>  => ? ∊ {0,4,24}
[3,2]
 => [2]
 => []
 =>  => ? ∊ {0,4,24}
[3,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[2,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[6]
 => []
 => ?
 => ? => ? ∊ {2,6,8,120}
[5,1]
 => [1]
 => []
 =>  => ? ∊ {2,6,8,120}
[4,2]
 => [2]
 => []
 =>  => ? ∊ {2,6,8,120}
[4,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[3,3]
 => [3]
 => []
 =>  => ? ∊ {2,6,8,120}
[3,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[2,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[7]
 => []
 => ?
 => ? => ? ∊ {0,0,6,720}
[6,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,6,720}
[5,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,6,720}
[5,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[4,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,6,720}
[4,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[3,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[3,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[8]
 => []
 => ?
 => ? => ? ∊ {0,4,8,48,5040}
[7,1]
 => [1]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[6,2]
 => [2]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[6,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[5,3]
 => [3]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[5,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[4,4]
 => [4]
 => []
 =>  => ? ∊ {0,4,8,48,5040}
[4,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[4,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[3,3,2]
 => [3,2]
 => [2]
 => 100 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => 110 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1010 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1110 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => 1100 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => 10110 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 0
[9]
 => []
 => ?
 => ? => ? ∊ {0,0,6,36,40320}
[8,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[7,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[7,1,1]
 => [1,1]
 => [1]
 => 10 => 0
[6,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[6,2,1]
 => [2,1]
 => [1]
 => 10 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => 110 => 0
[5,4]
 => [4]
 => []
 =>  => ? ∊ {0,0,6,36,40320}
[5,3,1]
 => [3,1]
 => [1]
 => 10 => 0
[5,2,2]
 => [2,2]
 => [2]
 => 100 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => 110 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1110 => 0
[4,4,1]
 => [4,1]
 => [1]
 => 10 => 0
[10]
 => []
 => ?
 => ? => ? ∊ {0,0,4,20,384,362880}
[9,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[8,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[7,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[6,4]
 => [4]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[5,5]
 => [5]
 => []
 =>  => ? ∊ {0,0,4,20,384,362880}
[11]
 => []
 => ?
 => ? => ? ∊ {0,0,0,0,0,10,3628800}
[10,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,0,0,0,10,3628800}
[9,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,0,0,0,10,3628800}
[8,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,0,0,0,10,3628800}
[7,4]
 => [4]
 => []
 =>  => ? ∊ {0,0,0,0,0,10,3628800}
[6,5]
 => [5]
 => []
 =>  => ? ∊ {0,0,0,0,0,10,3628800}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1111111110 => ? ∊ {0,0,0,0,0,10,3628800}
[12]
 => []
 => ?
 => ? => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[11,1]
 => [1]
 => []
 =>  => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[10,2]
 => [2]
 => []
 =>  => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[9,3]
 => [3]
 => []
 =>  => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[8,4]
 => [4]
 => []
 =>  => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[7,5]
 => [5]
 => []
 =>  => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[6,6]
 => [6]
 => []
 =>  => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1111111110 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => 11111111110 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}

Description

The length of the longest Yamanouchi prefix of a binary word. This is the largest index

$i$ such that in each of the prefixes

$w_1$ ,

$w_1w_2$ ,

$w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.

Matching statistic: St001695

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 81%●distinct values known / distinct values provided: 4%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {2,2}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {2,2}
[1,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {2,2,6}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {2,2,6}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {2,2,6}
[2,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,4,24}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,4,24}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,4,24}
[3,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {2,6,8,120}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {2,6,8,120}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {2,6,8,120}
[4,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {2,6,8,120}
[3,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,6,720}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,6,720}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,6,720}
[5,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,6,720}
[4,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,4,8,48,5040}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[6,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[5,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[4,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,6,36,40320}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[7,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[6,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[5,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [[1]]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,4,20,384,362880}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,10,3628800}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,0,0,10,3628800}
[12]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[6,6]
 => [6]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}

Description

The natural comajor index of a standard Young tableau. A natural descent of a standard tableau

$T$ is an entry

$i$ such that

$i+1$ appears in a higher row than

$i$ in English notation. The natural comajor index of a tableau of size

$n$ with natural descent set

$D$ is then

$\sum_{d\in D} n-d$ .

Matching statistic: St001698

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 81%●distinct values known / distinct values provided: 4%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {2,2}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {2,2}
[1,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {2,2,6}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {2,2,6}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {2,2,6}
[2,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,4,24}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,4,24}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,4,24}
[3,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {2,6,8,120}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {2,6,8,120}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {2,6,8,120}
[4,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {2,6,8,120}
[3,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,6,720}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,6,720}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,6,720}
[5,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,6,720}
[4,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,4,8,48,5040}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[6,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[5,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[4,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,6,36,40320}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[7,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[6,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[5,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [[1]]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,4,20,384,362880}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,10,3628800}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,0,0,10,3628800}
[12]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[6,6]
 => [6]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}

Description

The comajor index of a standard tableau minus the weighted size of its shape.

Matching statistic: St001699

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 81%●distinct values known / distinct values provided: 4%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {2,2}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {2,2}
[1,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {2,2,6}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {2,2,6}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {2,2,6}
[2,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,4,24}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,4,24}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,4,24}
[3,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {2,6,8,120}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {2,6,8,120}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {2,6,8,120}
[4,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {2,6,8,120}
[3,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,6,720}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,6,720}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,6,720}
[5,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,6,720}
[4,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,3],[2]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,4,8,48,5040}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[6,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[5,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[4,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,3],[2]]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,6,36,40320}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[7,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[6,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[5,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [[1]]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,4,20,384,362880}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,10,3628800}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,0,0,10,3628800}
[12]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[6,6]
 => [6]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}

Description

The major index of a standard tableau minus the weighted size of its shape.

Matching statistic: St001712

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 4% ●values known / values provided: 81%●distinct values known / distinct values provided: 4%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {2,2}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {2,2}
[1,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {2,2,6}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {2,2,6}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {2,2,6}
[2,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,4,24}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,4,24}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,4,24}
[3,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {2,6,8,120}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {2,6,8,120}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {2,6,8,120}
[4,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {2,6,8,120}
[3,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,6,720}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,6,720}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,6,720}
[5,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,6,720}
[4,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,4,8,48,5040}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[6,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[5,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[4,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,6,36,40320}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[7,1,1]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[6,2,1]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[5,3,1]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [[1]]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,4,20,384,362880}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,4,20,384,362880}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,10,3628800}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,10,3628800}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,0,0,10,3628800}
[12]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[6,6]
 => [6]
 => []
 => []
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? ∊ {0,0,0,4,12,64,324,3840,39916800}

Description

The number of natural descents of a standard Young tableau. A natural descent of a standard tableau

$T$ is an entry

$i$ such that

$i+1$ appears in a higher row than

$i$ in English notation.

Matching statistic: St001107

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 80%●distinct values known / distinct values provided: 4%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {2,2}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {2,2}
[1,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {2,2,6}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {2,2,6}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {2,2,6}
[2,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,4,24}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,4,24}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,4,24}
[3,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {2,6,8,120}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {2,6,8,120}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {2,6,8,120}
[4,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {2,6,8,120}
[3,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,6,720}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,6,720}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,6,720}
[5,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,6,720}
[4,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,4,8,48,5040}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[6,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[5,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,4,8,48,5040}
[4,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,6,36,40320}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[7,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[6,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,6,36,40320}
[5,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1,0,1,0]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,4,20,384,362880}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,4,20,384,362880}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,4,20,384,362880}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,4,20,384,362880}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,4,20,384,362880}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,4,20,384,362880}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,4,20,384,362880}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,10,3628800}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,10,3628800}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,10,3628800}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,10,3628800}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,10,3628800}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,10,3628800}
[2,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,10,3628800}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,10,3628800}
[12]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,4,12,64,324,3840,39916800}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,4,12,64,324,3840,39916800}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,4,12,64,324,3840,39916800}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,4,12,64,324,3840,39916800}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,4,12,64,324,3840,39916800}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,4,12,64,324,3840,39916800}
[6,6]
 => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,4,12,64,324,3840,39916800}

Description

The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.

The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1.