Your data matches 284 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000527
(load all 6 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => 1
[2,1] => ([],2)
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => 2
[2,3,1] => ([(1,2)],3)
 => 2
[3,1,2] => ([(1,2)],3)
 => 2
[3,2,1] => ([],3)
 => 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[2,3,4,1] => ([(1,2),(2,3)],4)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 2
[2,4,3,1] => ([(1,2),(1,3)],4)
 => 3
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(1,3),(2,3)],4)
 => 3
[3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => 3
[4,1,2,3] => ([(1,2),(2,3)],4)
 => 2
[4,1,3,2] => ([(1,2),(1,3)],4)
 => 3
[4,2,1,3] => ([(1,3),(2,3)],4)
 => 3
[4,2,3,1] => ([(2,3)],4)
 => 3
[4,3,1,2] => ([(2,3)],4)
 => 3
[4,3,2,1] => ([],4)
 => 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
Description
The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000147
(load all 9 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 1
[1,2] => [1,1]
 => 1
[2,1] => [2]
 => 2
[1,2,3] => [1,1,1]
 => 1
[1,3,2] => [2,1]
 => 2
[2,1,3] => [2,1]
 => 2
[2,3,1] => [2,1]
 => 2
[3,1,2] => [2,1]
 => 2
[3,2,1] => [3]
 => 3
[1,2,3,4] => [1,1,1,1]
 => 1
[1,2,4,3] => [2,1,1]
 => 2
[1,3,2,4] => [2,1,1]
 => 2
[1,3,4,2] => [2,1,1]
 => 2
[1,4,2,3] => [2,1,1]
 => 2
[1,4,3,2] => [3,1]
 => 3
[2,1,3,4] => [2,1,1]
 => 2
[2,1,4,3] => [2,2]
 => 2
[2,3,1,4] => [2,1,1]
 => 2
[2,3,4,1] => [2,1,1]
 => 2
[2,4,1,3] => [2,1,1]
 => 2
[2,4,3,1] => [3,1]
 => 3
[3,1,2,4] => [2,1,1]
 => 2
[3,1,4,2] => [2,2]
 => 2
[3,2,1,4] => [3,1]
 => 3
[3,2,4,1] => [3,1]
 => 3
[3,4,1,2] => [2,1,1]
 => 2
[3,4,2,1] => [3,1]
 => 3
[4,1,2,3] => [2,1,1]
 => 2
[4,1,3,2] => [3,1]
 => 3
[4,2,1,3] => [3,1]
 => 3
[4,2,3,1] => [3,1]
 => 3
[4,3,1,2] => [3,1]
 => 3
[4,3,2,1] => [4]
 => 4
[1,2,3,4,5] => [1,1,1,1,1]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => 2
[1,2,4,5,3] => [2,1,1,1]
 => 2
[1,2,5,3,4] => [2,1,1,1]
 => 2
[1,2,5,4,3] => [3,1,1]
 => 3
[1,3,2,4,5] => [2,1,1,1]
 => 2
[1,3,2,5,4] => [2,2,1]
 => 2
[1,3,4,2,5] => [2,1,1,1]
 => 2
[1,3,4,5,2] => [2,1,1,1]
 => 2
[1,3,5,2,4] => [2,1,1,1]
 => 2
[1,3,5,4,2] => [3,1,1]
 => 3
[1,4,2,3,5] => [2,1,1,1]
 => 2
[1,4,2,5,3] => [2,2,1]
 => 2
[1,4,3,2,5] => [3,1,1]
 => 3
[1,4,3,5,2] => [3,1,1]
 => 3
[1,4,5,2,3] => [2,1,1,1]
 => 2
[8,7,5,6,4,2,3,1] => ?
 => ? = 6
[8,5,7,6,4,1,3,2] => ?
 => ? = 6
[8,7,4,3,2,1,6,5] => ?
 => ? = 6
[8,7,4,6,5,1,3,2] => ?
 => ? = 6
Description
The largest part of an integer partition.
Matching statistic: St000010
(load all 7 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [1]
 => 1
[1,2] => [1,1]
 => [2]
 => 1
[2,1] => [2]
 => [1,1]
 => 2
[1,2,3] => [1,1,1]
 => [3]
 => 1
[1,3,2] => [2,1]
 => [2,1]
 => 2
[2,1,3] => [2,1]
 => [2,1]
 => 2
[2,3,1] => [2,1]
 => [2,1]
 => 2
[3,1,2] => [2,1]
 => [2,1]
 => 2
[3,2,1] => [3]
 => [1,1,1]
 => 3
[1,2,3,4] => [1,1,1,1]
 => [4]
 => 1
[1,2,4,3] => [2,1,1]
 => [3,1]
 => 2
[1,3,2,4] => [2,1,1]
 => [3,1]
 => 2
[1,3,4,2] => [2,1,1]
 => [3,1]
 => 2
[1,4,2,3] => [2,1,1]
 => [3,1]
 => 2
[1,4,3,2] => [3,1]
 => [2,1,1]
 => 3
[2,1,3,4] => [2,1,1]
 => [3,1]
 => 2
[2,1,4,3] => [2,2]
 => [2,2]
 => 2
[2,3,1,4] => [2,1,1]
 => [3,1]
 => 2
[2,3,4,1] => [2,1,1]
 => [3,1]
 => 2
[2,4,1,3] => [2,1,1]
 => [3,1]
 => 2
[2,4,3,1] => [3,1]
 => [2,1,1]
 => 3
[3,1,2,4] => [2,1,1]
 => [3,1]
 => 2
[3,1,4,2] => [2,2]
 => [2,2]
 => 2
[3,2,1,4] => [3,1]
 => [2,1,1]
 => 3
[3,2,4,1] => [3,1]
 => [2,1,1]
 => 3
[3,4,1,2] => [2,1,1]
 => [3,1]
 => 2
[3,4,2,1] => [3,1]
 => [2,1,1]
 => 3
[4,1,2,3] => [2,1,1]
 => [3,1]
 => 2
[4,1,3,2] => [3,1]
 => [2,1,1]
 => 3
[4,2,1,3] => [3,1]
 => [2,1,1]
 => 3
[4,2,3,1] => [3,1]
 => [2,1,1]
 => 3
[4,3,1,2] => [3,1]
 => [2,1,1]
 => 3
[4,3,2,1] => [4]
 => [1,1,1,1]
 => 4
[1,2,3,4,5] => [1,1,1,1,1]
 => [5]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [4,1]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [4,1]
 => 2
[1,2,4,5,3] => [2,1,1,1]
 => [4,1]
 => 2
[1,2,5,3,4] => [2,1,1,1]
 => [4,1]
 => 2
[1,2,5,4,3] => [3,1,1]
 => [3,1,1]
 => 3
[1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [3,2]
 => 2
[1,3,4,2,5] => [2,1,1,1]
 => [4,1]
 => 2
[1,3,4,5,2] => [2,1,1,1]
 => [4,1]
 => 2
[1,3,5,2,4] => [2,1,1,1]
 => [4,1]
 => 2
[1,3,5,4,2] => [3,1,1]
 => [3,1,1]
 => 3
[1,4,2,3,5] => [2,1,1,1]
 => [4,1]
 => 2
[1,4,2,5,3] => [2,2,1]
 => [3,2]
 => 2
[1,4,3,2,5] => [3,1,1]
 => [3,1,1]
 => 3
[1,4,3,5,2] => [3,1,1]
 => [3,1,1]
 => 3
[1,4,5,2,3] => [2,1,1,1]
 => [4,1]
 => 2
[8,7,5,6,4,2,3,1] => ?
 => ?
 => ? = 6
[8,5,7,6,4,1,3,2] => ?
 => ?
 => ? = 6
[8,7,4,3,2,1,6,5] => ?
 => ?
 => ? = 6
[8,7,4,6,5,1,3,2] => ?
 => ?
 => ? = 6
Description
The length of the partition.
Matching statistic: St000288
(load all 21 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 10 => 01 => 1
[1,2] => [1,1]
 => 110 => 001 => 1
[2,1] => [2]
 => 100 => 011 => 2
[1,2,3] => [1,1,1]
 => 1110 => 0001 => 1
[1,3,2] => [2,1]
 => 1010 => 0101 => 2
[2,1,3] => [2,1]
 => 1010 => 0101 => 2
[2,3,1] => [2,1]
 => 1010 => 0101 => 2
[3,1,2] => [2,1]
 => 1010 => 0101 => 2
[3,2,1] => [3]
 => 1000 => 0111 => 3
[1,2,3,4] => [1,1,1,1]
 => 11110 => 00001 => 1
[1,2,4,3] => [2,1,1]
 => 10110 => 01001 => 2
[1,3,2,4] => [2,1,1]
 => 10110 => 01001 => 2
[1,3,4,2] => [2,1,1]
 => 10110 => 01001 => 2
[1,4,2,3] => [2,1,1]
 => 10110 => 01001 => 2
[1,4,3,2] => [3,1]
 => 10010 => 01101 => 3
[2,1,3,4] => [2,1,1]
 => 10110 => 01001 => 2
[2,1,4,3] => [2,2]
 => 1100 => 0011 => 2
[2,3,1,4] => [2,1,1]
 => 10110 => 01001 => 2
[2,3,4,1] => [2,1,1]
 => 10110 => 01001 => 2
[2,4,1,3] => [2,1,1]
 => 10110 => 01001 => 2
[2,4,3,1] => [3,1]
 => 10010 => 01101 => 3
[3,1,2,4] => [2,1,1]
 => 10110 => 01001 => 2
[3,1,4,2] => [2,2]
 => 1100 => 0011 => 2
[3,2,1,4] => [3,1]
 => 10010 => 01101 => 3
[3,2,4,1] => [3,1]
 => 10010 => 01101 => 3
[3,4,1,2] => [2,1,1]
 => 10110 => 01001 => 2
[3,4,2,1] => [3,1]
 => 10010 => 01101 => 3
[4,1,2,3] => [2,1,1]
 => 10110 => 01001 => 2
[4,1,3,2] => [3,1]
 => 10010 => 01101 => 3
[4,2,1,3] => [3,1]
 => 10010 => 01101 => 3
[4,2,3,1] => [3,1]
 => 10010 => 01101 => 3
[4,3,1,2] => [3,1]
 => 10010 => 01101 => 3
[4,3,2,1] => [4]
 => 10000 => 01111 => 4
[1,2,3,4,5] => [1,1,1,1,1]
 => 111110 => 000001 => 1
[1,2,3,5,4] => [2,1,1,1]
 => 101110 => 010001 => 2
[1,2,4,3,5] => [2,1,1,1]
 => 101110 => 010001 => 2
[1,2,4,5,3] => [2,1,1,1]
 => 101110 => 010001 => 2
[1,2,5,3,4] => [2,1,1,1]
 => 101110 => 010001 => 2
[1,2,5,4,3] => [3,1,1]
 => 100110 => 011001 => 3
[1,3,2,4,5] => [2,1,1,1]
 => 101110 => 010001 => 2
[1,3,2,5,4] => [2,2,1]
 => 11010 => 00101 => 2
[1,3,4,2,5] => [2,1,1,1]
 => 101110 => 010001 => 2
[1,3,4,5,2] => [2,1,1,1]
 => 101110 => 010001 => 2
[1,3,5,2,4] => [2,1,1,1]
 => 101110 => 010001 => 2
[1,3,5,4,2] => [3,1,1]
 => 100110 => 011001 => 3
[1,4,2,3,5] => [2,1,1,1]
 => 101110 => 010001 => 2
[1,4,2,5,3] => [2,2,1]
 => 11010 => 00101 => 2
[1,4,3,2,5] => [3,1,1]
 => 100110 => 011001 => 3
[1,4,3,5,2] => [3,1,1]
 => 100110 => 011001 => 3
[1,4,5,2,3] => [2,1,1,1]
 => 101110 => 010001 => 2
[8,7,5,6,4,2,3,1] => ?
 => ? => ? => ? = 6
[8,5,7,6,4,1,3,2] => ?
 => ? => ? => ? = 6
[8,7,4,3,2,1,6,5] => ?
 => ? => ? => ? = 6
[8,7,4,6,5,1,3,2] => ?
 => ? => ? => ? = 6
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000378
(load all 3 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [1]
 => [1]
 => 1
[1,2] => [1,1]
 => [2]
 => [1,1]
 => 1
[2,1] => [2]
 => [1,1]
 => [2]
 => 2
[1,2,3] => [1,1,1]
 => [3]
 => [1,1,1]
 => 1
[1,3,2] => [2,1]
 => [2,1]
 => [3]
 => 2
[2,1,3] => [2,1]
 => [2,1]
 => [3]
 => 2
[2,3,1] => [2,1]
 => [2,1]
 => [3]
 => 2
[3,1,2] => [2,1]
 => [2,1]
 => [3]
 => 2
[3,2,1] => [3]
 => [1,1,1]
 => [2,1]
 => 3
[1,2,3,4] => [1,1,1,1]
 => [4]
 => [1,1,1,1]
 => 1
[1,2,4,3] => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[1,3,2,4] => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[1,3,4,2] => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[1,4,2,3] => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[1,4,3,2] => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
[2,1,3,4] => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[2,1,4,3] => [2,2]
 => [2,2]
 => [4]
 => 2
[2,3,1,4] => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[2,3,4,1] => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[2,4,1,3] => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[2,4,3,1] => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
[3,1,2,4] => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[3,1,4,2] => [2,2]
 => [2,2]
 => [4]
 => 2
[3,2,1,4] => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
[3,2,4,1] => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
[3,4,1,2] => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[3,4,2,1] => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
[4,1,2,3] => [2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[4,1,3,2] => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
[4,2,1,3] => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
[4,2,3,1] => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
[4,3,1,2] => [3,1]
 => [2,1,1]
 => [2,2]
 => 3
[4,3,2,1] => [4]
 => [1,1,1,1]
 => [3,1]
 => 4
[1,2,3,4,5] => [1,1,1,1,1]
 => [5]
 => [1,1,1,1,1]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
[1,2,4,5,3] => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
[1,2,5,3,4] => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
[1,2,5,4,3] => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
[1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [3,2]
 => [5]
 => 2
[1,3,4,2,5] => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
[1,3,4,5,2] => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
[1,3,5,2,4] => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
[1,3,5,4,2] => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
[1,4,2,3,5] => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
[1,4,2,5,3] => [2,2,1]
 => [3,2]
 => [5]
 => 2
[1,4,3,2,5] => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
[1,4,3,5,2] => [3,1,1]
 => [3,1,1]
 => [4,1]
 => 3
[1,4,5,2,3] => [2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 2
[8,7,5,6,4,2,3,1] => ?
 => ?
 => ?
 => ? = 6
[8,5,7,6,4,1,3,2] => ?
 => ?
 => ?
 => ? = 6
[8,7,4,3,2,1,6,5] => ?
 => ?
 => ?
 => ? = 6
[8,7,4,6,5,1,3,2] => ?
 => ?
 => ?
 => ? = 6
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000734
(load all 3 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => 1
[1,2] => [1,1]
 => [[1],[2]]
 => 1
[2,1] => [2]
 => [[1,2]]
 => 2
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => 1
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => 2
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => 2
[2,3,1] => [2,1]
 => [[1,2],[3]]
 => 2
[3,1,2] => [2,1]
 => [[1,2],[3]]
 => 2
[3,2,1] => [3]
 => [[1,2,3]]
 => 3
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
[1,2,4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[1,3,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[1,3,4,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[1,4,2,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[1,4,3,2] => [3,1]
 => [[1,2,3],[4]]
 => 3
[2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => 2
[2,3,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[2,3,4,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[2,4,1,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[2,4,3,1] => [3,1]
 => [[1,2,3],[4]]
 => 3
[3,1,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => 2
[3,2,1,4] => [3,1]
 => [[1,2,3],[4]]
 => 3
[3,2,4,1] => [3,1]
 => [[1,2,3],[4]]
 => 3
[3,4,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[3,4,2,1] => [3,1]
 => [[1,2,3],[4]]
 => 3
[4,1,2,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,1,3,2] => [3,1]
 => [[1,2,3],[4]]
 => 3
[4,2,1,3] => [3,1]
 => [[1,2,3],[4]]
 => 3
[4,2,3,1] => [3,1]
 => [[1,2,3],[4]]
 => 3
[4,3,1,2] => [3,1]
 => [[1,2,3],[4]]
 => 3
[4,3,2,1] => [4]
 => [[1,2,3,4]]
 => 4
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,2,4,5,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,2,5,3,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,3,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
[1,3,4,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,3,4,5,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,3,5,2,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,2,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,4,2,5,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
[1,4,3,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,5,2,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[8,7,5,6,4,2,3,1] => ?
 => ?
 => ? = 6
[8,5,7,6,4,1,3,2] => ?
 => ?
 => ? = 6
[8,7,4,3,2,1,6,5] => ?
 => ?
 => ? = 6
[8,7,4,6,5,1,3,2] => ?
 => ?
 => ? = 6
[6,5,7,4,8,3,9,2,10,1,11] => [6,1,1,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
 => ? = 6
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000013
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [1,0]
 => [1,0]
 => 1
[1,2] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[1,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[2,1,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[2,3,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[3,1,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[3,2,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,3,4,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,4,2,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,4,3,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[2,1,4,3] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[2,3,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[2,3,4,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[2,4,1,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[2,4,3,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[3,1,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[3,1,4,2] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[3,2,1,4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[3,2,4,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[3,4,1,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[3,4,2,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[4,1,2,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[4,1,3,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[4,2,1,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[4,2,3,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[4,3,1,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[4,3,2,1] => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,2,4,5,3] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,2,5,3,4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,2,5,4,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,3,2,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,3,4,2,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,3,4,5,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,3,5,2,4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,3,5,4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,4,2,3,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,4,2,5,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,4,3,2,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,4,3,5,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,4,5,2,3] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[8,7,5,6,4,2,3,1] => ?
 => ?
 => ?
 => ? = 6
[8,5,7,6,4,1,3,2] => ?
 => ?
 => ?
 => ? = 6
[8,7,4,3,2,1,6,5] => ?
 => ?
 => ?
 => ? = 6
[8,7,4,6,5,1,3,2] => ?
 => ?
 => ?
 => ? = 6
[6,5,7,4,8,3,9,2,10,1,11] => [6,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000141
(load all 3 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [1,0,1,0]
 => [2,1] => 1
[1,2] => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[2,1] => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[1,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2
[2,1,3] => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2
[2,3,1] => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2
[3,1,2] => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 2
[3,2,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[1,3,4,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[1,4,2,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[1,4,3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[2,1,4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[2,3,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[2,3,4,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[2,4,1,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[2,4,3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[3,1,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[3,1,4,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[3,2,1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[3,2,4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[3,4,1,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[3,4,2,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[4,1,2,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[4,1,3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[4,2,1,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[4,2,3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[4,3,1,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[4,3,2,1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 4
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
[1,2,4,3,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
[1,2,4,5,3] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
[1,2,5,3,4] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
[1,2,5,4,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[1,3,2,4,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
[1,3,2,5,4] => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2
[1,3,4,2,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
[1,3,4,5,2] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
[1,3,5,2,4] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
[1,3,5,4,2] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[1,4,2,3,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
[1,4,2,5,3] => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2
[1,4,3,2,5] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[1,4,3,5,2] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 3
[1,4,5,2,3] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 2
[8,7,5,6,4,2,3,1] => ?
 => ?
 => ? => ? = 6
[8,5,7,6,4,1,3,2] => ?
 => ?
 => ? => ? = 6
[8,7,4,3,2,1,6,5] => ?
 => ?
 => ? => ? = 6
[8,7,4,6,5,1,3,2] => ?
 => ?
 => ? => ? = 6
[6,5,7,4,8,3,9,2,10,1,11] => [6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,6,1] => ? = 6
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [1] => 1
[1,2] => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[2,1] => [2]
 => [[1,2]]
 => [2] => 2
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => [2,1] => 2
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => [2,1] => 2
[2,3,1] => [2,1]
 => [[1,2],[3]]
 => [2,1] => 2
[3,1,2] => [2,1]
 => [[1,2],[3]]
 => [2,1] => 2
[3,2,1] => [3]
 => [[1,2,3]]
 => [3] => 3
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[1,2,4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[1,3,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[1,3,4,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[1,4,2,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[1,4,3,2] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 3
[2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[2,3,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[2,3,4,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[2,4,1,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[2,4,3,1] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 3
[3,1,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[3,2,1,4] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 3
[3,2,4,1] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 3
[3,4,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[3,4,2,1] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 3
[4,1,2,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[4,1,3,2] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 3
[4,2,1,3] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 3
[4,2,3,1] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 3
[4,3,1,2] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 3
[4,3,2,1] => [4]
 => [[1,2,3,4]]
 => [4] => 4
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
[1,2,3,5,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
[1,2,4,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
[1,2,4,5,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
[1,2,5,3,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
[1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 3
[1,3,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
[1,3,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2
[1,3,4,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
[1,3,4,5,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
[1,3,5,2,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 3
[1,4,2,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
[1,4,2,5,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2
[1,4,3,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 3
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 3
[1,4,5,2,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
[8,7,5,6,4,2,3,1] => ?
 => ?
 => ? => ? = 6
[8,5,7,6,4,1,3,2] => ?
 => ?
 => ? => ? = 6
[8,7,4,3,2,1,6,5] => ?
 => ?
 => ? => ? = 6
[8,7,4,6,5,1,3,2] => ?
 => ?
 => ? => ? = 6
[6,5,7,4,8,3,9,2,10,1,11] => [6,1,1,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
 => ? => ? = 6
Description
The first part of an integer composition.
Matching statistic: St000733
(load all 5 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [[1]]
 => 1
[1,2] => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
[2,1] => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[2,3,1] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[3,1,2] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[3,2,1] => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
[1,3,4,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
[1,4,2,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
[1,4,3,2] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
[2,1,3,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[2,3,1,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
[2,3,4,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
[2,4,1,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
[2,4,3,1] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
[3,1,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[3,2,1,4] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
[3,2,4,1] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
[3,4,1,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
[3,4,2,1] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
[4,1,2,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
[4,1,3,2] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
[4,2,1,3] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
[4,2,3,1] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
[4,3,1,2] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
[4,3,2,1] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
[1,2,4,5,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
[1,2,5,3,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
[1,2,5,4,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[1,3,2,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 2
[1,3,4,2,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
[1,3,4,5,2] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
[1,3,5,2,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
[1,3,5,4,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,2,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
[1,4,2,5,3] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 2
[1,4,3,2,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,3,5,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,5,2,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 2
[8,7,5,6,4,2,3,1] => ?
 => ?
 => ?
 => ? = 6
[8,5,7,6,4,1,3,2] => ?
 => ?
 => ?
 => ? = 6
[8,7,4,3,2,1,6,5] => ?
 => ?
 => ?
 => ? = 6
[8,7,4,6,5,1,3,2] => ?
 => ?
 => ?
 => ? = 6
[6,5,7,4,8,3,9,2,10,1,11] => [6,1,1,1,1,1]
 => [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
 => ?
 => ? = 6
Description
The row containing the largest entry of a standard tableau.
The following 274 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000054The first entry of the permutation. St000157The number of descents of a standard tableau. St000326The position of the first one in a binary word after appending a 1 at the end. St000738The first entry in the last row of a standard tableau. St000381The largest part of an integer composition. St000806The semiperimeter of the associated bargraph. St000007The number of saliances of the permutation. St000507The number of ascents of a standard tableau. St000546The number of global descents of a permutation. St000808The number of up steps of the associated bargraph. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001372The length of a longest cyclic run of ones of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000519The largest length of a factor maximising the subword complexity. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000691The number of changes of a binary word. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001389The number of partitions of the same length below the given integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000439The position of the first down step of a Dyck path. St000809The reduced reflection length of the permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000025The number of initial rises of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000504The cardinality of the first block of a set partition. St001062The maximal size of a block of a set partition. St000383The last part of an integer composition. St001029The size of the core of a graph. St000925The number of topologically connected components of a set partition. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St000823The number of unsplittable factors of the set partition. St000503The maximal difference between two elements in a common block. St001497The position of the largest weak excedence of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000105The number of blocks in the set partition. St001777The number of weak descents in an integer composition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000093The cardinality of a maximal independent set of vertices of a graph. St000006The dinv of a Dyck path. St000015The number of peaks of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000216The absolute length of a permutation. St000632The jump number of the poset. St000840The number of closers smaller than the largest opener in a perfect matching. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000740The last entry of a permutation. St000662The staircase size of the code of a permutation. St000528The height of a poset. St000308The height of the tree associated to a permutation. St000741The Colin de Verdière graph invariant. St000031The number of cycles in the cycle decomposition of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000083The number of left oriented leafs of a binary tree except the first one. St000068The number of minimal elements in a poset. St000245The number of ascents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000470The number of runs in a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000703The number of deficiencies of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001427The number of descents of a signed permutation. St000354The number of recoils of a permutation. St000542The number of left-to-right-minima of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000062The length of the longest increasing subsequence of the permutation. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000325The width of the tree associated to a permutation. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000926The clique-coclique number of a graph. St000991The number of right-to-left minima of a permutation. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000021The number of descents of a permutation. St000080The rank of the poset. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000454The largest eigenvalue of a graph if it is integral. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000314The number of left-to-right-maxima of a permutation. St000315The number of isolated vertices of a graph. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000544The cop number of a graph. St000553The number of blocks of a graph. St000667The greatest common divisor of the parts of the partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001286The annihilation number of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001691The number of kings in a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St000008The major index of the composition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000145The Dyson rank of a partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000831The number of indices that are either descents or recoils. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001340The cardinality of a minimal non-edge isolating set of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001814The number of partitions interlacing the given partition. St001812The biclique partition number of a graph. St000061The number of nodes on the left branch of a binary tree. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001834The number of non-isomorphic minors of a graph. St001706The number of closed sets in a graph. St000477The weight of a partition according to Alladi. St000770The major index of an integer partition when read from bottom to top. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001674The number of vertices of the largest induced star graph in the graph. St001323The independence gap of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001651The Frankl number of a lattice. St001875The number of simple modules with projective dimension at most 1. St001720The minimal length of a chain of small intervals in a lattice. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001946The number of descents in a parking function. St001624The breadth of a lattice.