Identifier
-
Mp00045:
Integer partitions
—reading tableau⟶
Standard tableaux
St000330: Standard tableaux ⟶ ℤ
Values
=>
Cc0002;cc-rep-0
Cc0007;cc-rep
[1]=>[[1]]=>0
[2]=>[[1,2]]=>0
[1,1]=>[[1],[2]]=>1
[3]=>[[1,2,3]]=>0
[2,1]=>[[1,3],[2]]=>1
[1,1,1]=>[[1],[2],[3]]=>3
[4]=>[[1,2,3,4]]=>0
[3,1]=>[[1,3,4],[2]]=>1
[2,2]=>[[1,2],[3,4]]=>2
[2,1,1]=>[[1,4],[2],[3]]=>3
[1,1,1,1]=>[[1],[2],[3],[4]]=>6
[5]=>[[1,2,3,4,5]]=>0
[4,1]=>[[1,3,4,5],[2]]=>1
[3,2]=>[[1,2,5],[3,4]]=>2
[3,1,1]=>[[1,4,5],[2],[3]]=>3
[2,2,1]=>[[1,3],[2,5],[4]]=>4
[2,1,1,1]=>[[1,5],[2],[3],[4]]=>6
[1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>10
[6]=>[[1,2,3,4,5,6]]=>0
[5,1]=>[[1,3,4,5,6],[2]]=>1
[4,2]=>[[1,2,5,6],[3,4]]=>2
[4,1,1]=>[[1,4,5,6],[2],[3]]=>3
[3,3]=>[[1,2,3],[4,5,6]]=>3
[3,2,1]=>[[1,3,6],[2,5],[4]]=>4
[3,1,1,1]=>[[1,5,6],[2],[3],[4]]=>6
[2,2,2]=>[[1,2],[3,4],[5,6]]=>6
[2,2,1,1]=>[[1,4],[2,6],[3],[5]]=>7
[2,1,1,1,1]=>[[1,6],[2],[3],[4],[5]]=>10
[1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>15
[7]=>[[1,2,3,4,5,6,7]]=>0
[6,1]=>[[1,3,4,5,6,7],[2]]=>1
[5,2]=>[[1,2,5,6,7],[3,4]]=>2
[5,1,1]=>[[1,4,5,6,7],[2],[3]]=>3
[4,3]=>[[1,2,3,7],[4,5,6]]=>3
[4,2,1]=>[[1,3,6,7],[2,5],[4]]=>4
[4,1,1,1]=>[[1,5,6,7],[2],[3],[4]]=>6
[3,3,1]=>[[1,3,4],[2,6,7],[5]]=>5
[3,2,2]=>[[1,2,7],[3,4],[5,6]]=>6
[3,2,1,1]=>[[1,4,7],[2,6],[3],[5]]=>7
[3,1,1,1,1]=>[[1,6,7],[2],[3],[4],[5]]=>10
[2,2,2,1]=>[[1,3],[2,5],[4,7],[6]]=>9
[2,2,1,1,1]=>[[1,5],[2,7],[3],[4],[6]]=>11
[2,1,1,1,1,1]=>[[1,7],[2],[3],[4],[5],[6]]=>15
[1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>21
[8]=>[[1,2,3,4,5,6,7,8]]=>0
[7,1]=>[[1,3,4,5,6,7,8],[2]]=>1
[6,2]=>[[1,2,5,6,7,8],[3,4]]=>2
[6,1,1]=>[[1,4,5,6,7,8],[2],[3]]=>3
[5,3]=>[[1,2,3,7,8],[4,5,6]]=>3
[5,2,1]=>[[1,3,6,7,8],[2,5],[4]]=>4
[5,1,1,1]=>[[1,5,6,7,8],[2],[3],[4]]=>6
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>4
[4,3,1]=>[[1,3,4,8],[2,6,7],[5]]=>5
[4,2,2]=>[[1,2,7,8],[3,4],[5,6]]=>6
[4,2,1,1]=>[[1,4,7,8],[2,6],[3],[5]]=>7
[4,1,1,1,1]=>[[1,6,7,8],[2],[3],[4],[5]]=>10
[3,3,2]=>[[1,2,5],[3,4,8],[6,7]]=>7
[3,3,1,1]=>[[1,4,5],[2,7,8],[3],[6]]=>8
[3,2,2,1]=>[[1,3,8],[2,5],[4,7],[6]]=>9
[3,2,1,1,1]=>[[1,5,8],[2,7],[3],[4],[6]]=>11
[3,1,1,1,1,1]=>[[1,7,8],[2],[3],[4],[5],[6]]=>15
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>12
[2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5],[7]]=>13
[2,2,1,1,1,1]=>[[1,6],[2,8],[3],[4],[5],[7]]=>16
[2,1,1,1,1,1,1]=>[[1,8],[2],[3],[4],[5],[6],[7]]=>21
[1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>28
[9]=>[[1,2,3,4,5,6,7,8,9]]=>0
[8,1]=>[[1,3,4,5,6,7,8,9],[2]]=>1
[7,2]=>[[1,2,5,6,7,8,9],[3,4]]=>2
[7,1,1]=>[[1,4,5,6,7,8,9],[2],[3]]=>3
[6,3]=>[[1,2,3,7,8,9],[4,5,6]]=>3
[6,2,1]=>[[1,3,6,7,8,9],[2,5],[4]]=>4
[6,1,1,1]=>[[1,5,6,7,8,9],[2],[3],[4]]=>6
[5,4]=>[[1,2,3,4,9],[5,6,7,8]]=>4
[5,3,1]=>[[1,3,4,8,9],[2,6,7],[5]]=>5
[5,2,2]=>[[1,2,7,8,9],[3,4],[5,6]]=>6
[5,2,1,1]=>[[1,4,7,8,9],[2,6],[3],[5]]=>7
[5,1,1,1,1]=>[[1,6,7,8,9],[2],[3],[4],[5]]=>10
[4,4,1]=>[[1,3,4,5],[2,7,8,9],[6]]=>6
[4,3,2]=>[[1,2,5,9],[3,4,8],[6,7]]=>7
[4,3,1,1]=>[[1,4,5,9],[2,7,8],[3],[6]]=>8
[4,2,2,1]=>[[1,3,8,9],[2,5],[4,7],[6]]=>9
[4,2,1,1,1]=>[[1,5,8,9],[2,7],[3],[4],[6]]=>11
[4,1,1,1,1,1]=>[[1,7,8,9],[2],[3],[4],[5],[6]]=>15
[3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>9
[3,3,2,1]=>[[1,3,6],[2,5,9],[4,8],[7]]=>10
[3,3,1,1,1]=>[[1,5,6],[2,8,9],[3],[4],[7]]=>12
[3,2,2,2]=>[[1,2,9],[3,4],[5,6],[7,8]]=>12
[3,2,2,1,1]=>[[1,4,9],[2,6],[3,8],[5],[7]]=>13
[3,2,1,1,1,1]=>[[1,6,9],[2,8],[3],[4],[5],[7]]=>16
[3,1,1,1,1,1,1]=>[[1,8,9],[2],[3],[4],[5],[6],[7]]=>21
[2,2,2,2,1]=>[[1,3],[2,5],[4,7],[6,9],[8]]=>16
[2,2,2,1,1,1]=>[[1,5],[2,7],[3,9],[4],[6],[8]]=>18
[2,2,1,1,1,1,1]=>[[1,7],[2,9],[3],[4],[5],[6],[8]]=>22
[2,1,1,1,1,1,1,1]=>[[1,9],[2],[3],[4],[5],[6],[7],[8]]=>28
[1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9]]=>36
[10]=>[[1,2,3,4,5,6,7,8,9,10]]=>0
[9,1]=>[[1,3,4,5,6,7,8,9,10],[2]]=>1
[8,2]=>[[1,2,5,6,7,8,9,10],[3,4]]=>2
[8,1,1]=>[[1,4,5,6,7,8,9,10],[2],[3]]=>3
[7,3]=>[[1,2,3,7,8,9,10],[4,5,6]]=>3
[7,2,1]=>[[1,3,6,7,8,9,10],[2,5],[4]]=>4
[7,1,1,1]=>[[1,5,6,7,8,9,10],[2],[3],[4]]=>6
[6,4]=>[[1,2,3,4,9,10],[5,6,7,8]]=>4
[6,3,1]=>[[1,3,4,8,9,10],[2,6,7],[5]]=>5
[6,2,2]=>[[1,2,7,8,9,10],[3,4],[5,6]]=>6
[6,2,1,1]=>[[1,4,7,8,9,10],[2,6],[3],[5]]=>7
[6,1,1,1,1]=>[[1,6,7,8,9,10],[2],[3],[4],[5]]=>10
[5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>5
[5,4,1]=>[[1,3,4,5,10],[2,7,8,9],[6]]=>6
[5,3,2]=>[[1,2,5,9,10],[3,4,8],[6,7]]=>7
[5,3,1,1]=>[[1,4,5,9,10],[2,7,8],[3],[6]]=>8
[5,2,2,1]=>[[1,3,8,9,10],[2,5],[4,7],[6]]=>9
[5,2,1,1,1]=>[[1,5,8,9,10],[2,7],[3],[4],[6]]=>11
[5,1,1,1,1,1]=>[[1,7,8,9,10],[2],[3],[4],[5],[6]]=>15
[4,4,2]=>[[1,2,5,6],[3,4,9,10],[7,8]]=>8
[4,4,1,1]=>[[1,4,5,6],[2,8,9,10],[3],[7]]=>9
[4,3,3]=>[[1,2,3,10],[4,5,6],[7,8,9]]=>9
[4,3,2,1]=>[[1,3,6,10],[2,5,9],[4,8],[7]]=>10
[4,3,1,1,1]=>[[1,5,6,10],[2,8,9],[3],[4],[7]]=>12
[4,2,2,2]=>[[1,2,9,10],[3,4],[5,6],[7,8]]=>12
[4,2,2,1,1]=>[[1,4,9,10],[2,6],[3,8],[5],[7]]=>13
[4,2,1,1,1,1]=>[[1,6,9,10],[2,8],[3],[4],[5],[7]]=>16
[4,1,1,1,1,1,1]=>[[1,8,9,10],[2],[3],[4],[5],[6],[7]]=>21
[3,3,3,1]=>[[1,3,4],[2,6,7],[5,9,10],[8]]=>12
[3,3,2,2]=>[[1,2,7],[3,4,10],[5,6],[8,9]]=>13
[3,3,2,1,1]=>[[1,4,7],[2,6,10],[3,9],[5],[8]]=>14
[3,3,1,1,1,1]=>[[1,6,7],[2,9,10],[3],[4],[5],[8]]=>17
[3,2,2,2,1]=>[[1,3,10],[2,5],[4,7],[6,9],[8]]=>16
[3,2,2,1,1,1]=>[[1,5,10],[2,7],[3,9],[4],[6],[8]]=>18
[3,2,1,1,1,1,1]=>[[1,7,10],[2,9],[3],[4],[5],[6],[8]]=>22
[3,1,1,1,1,1,1,1]=>[[1,9,10],[2],[3],[4],[5],[6],[7],[8]]=>28
[2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>20
[2,2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5,10],[7],[9]]=>21
[2,2,2,1,1,1,1]=>[[1,6],[2,8],[3,10],[4],[5],[7],[9]]=>24
[2,2,1,1,1,1,1,1]=>[[1,8],[2,10],[3],[4],[5],[6],[7],[9]]=>29
[2,1,1,1,1,1,1,1,1]=>[[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]=>36
[1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]=>45
[5,4,2]=>[[1,2,5,6,11],[3,4,9,10],[7,8]]=>8
[5,4,1,1]=>[[1,4,5,6,11],[2,8,9,10],[3],[7]]=>9
[5,3,3]=>[[1,2,3,10,11],[4,5,6],[7,8,9]]=>9
[5,3,2,1]=>[[1,3,6,10,11],[2,5,9],[4,8],[7]]=>10
[5,3,1,1,1]=>[[1,5,6,10,11],[2,8,9],[3],[4],[7]]=>12
[5,2,2,2]=>[[1,2,9,10,11],[3,4],[5,6],[7,8]]=>12
[5,2,2,1,1]=>[[1,4,9,10,11],[2,6],[3,8],[5],[7]]=>13
[4,4,3]=>[[1,2,3,7],[4,5,6,11],[8,9,10]]=>10
[4,4,2,1]=>[[1,3,6,7],[2,5,10,11],[4,9],[8]]=>11
[4,4,1,1,1]=>[[1,5,6,7],[2,9,10,11],[3],[4],[8]]=>13
[4,3,3,1]=>[[1,3,4,11],[2,6,7],[5,9,10],[8]]=>12
[4,3,2,2]=>[[1,2,7,11],[3,4,10],[5,6],[8,9]]=>13
[4,3,2,1,1]=>[[1,4,7,11],[2,6,10],[3,9],[5],[8]]=>14
[4,2,2,2,1]=>[[1,3,10,11],[2,5],[4,7],[6,9],[8]]=>16
[3,3,3,2]=>[[1,2,5],[3,4,8],[6,7,11],[9,10]]=>15
[3,3,3,1,1]=>[[1,4,5],[2,7,8],[3,10,11],[6],[9]]=>16
[3,3,2,2,1]=>[[1,3,8],[2,5,11],[4,7],[6,10],[9]]=>17
[12]=>[[1,2,3,4,5,6,7,8,9,10,11,12]]=>0
[6,6]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>6
[6,4,2]=>[[1,2,5,6,11,12],[3,4,9,10],[7,8]]=>8
[5,4,3]=>[[1,2,3,7,12],[4,5,6,11],[8,9,10]]=>10
[5,4,2,1]=>[[1,3,6,7,12],[2,5,10,11],[4,9],[8]]=>11
[5,4,1,1,1]=>[[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]=>13
[5,3,3,1]=>[[1,3,4,11,12],[2,6,7],[5,9,10],[8]]=>12
[5,3,2,2]=>[[1,2,7,11,12],[3,4,10],[5,6],[8,9]]=>13
[5,3,2,1,1]=>[[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]=>14
[5,2,2,2,1]=>[[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]=>16
[4,4,3,1]=>[[1,3,4,8],[2,6,7,12],[5,10,11],[9]]=>13
[4,4,2,2]=>[[1,2,7,8],[3,4,11,12],[5,6],[9,10]]=>14
[4,4,2,1,1]=>[[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]=>15
[4,3,3,2]=>[[1,2,5,12],[3,4,8],[6,7,11],[9,10]]=>15
[4,3,3,1,1]=>[[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]=>16
[4,3,2,2,1]=>[[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]=>17
[3,3,3,2,1]=>[[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]=>19
[3,3,2,2,1,1]=>[[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]=>22
[2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]=>30
[5,4,3,1]=>[[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]]=>13
[5,4,2,2]=>[[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]=>14
[5,4,2,1,1]=>[[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]=>15
[5,3,3,2]=>[[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]=>15
[5,3,3,1,1]=>[[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]=>16
[5,3,2,2,1]=>[[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]=>17
[4,4,3,2]=>[[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]=>16
[4,4,3,1,1]=>[[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]=>17
[4,4,2,2,1]=>[[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]=>18
[4,3,3,2,1]=>[[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]=>19
[5,4,3,2]=>[[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]=>16
[5,4,3,1,1]=>[[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]=>17
[5,4,2,2,1]=>[[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]=>18
[5,3,3,2,1]=>[[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]=>19
[4,4,3,2,1]=>[[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]=>20
[5,4,3,2,1]=>[[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]=>20
[]=>[]=>0
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Description
The (standard) major index of a standard tableau.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Map
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
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