- FindStat

Your data matches 178 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000288
(load all 35 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => 0 => 0
[1,1,0,0]
 => [2,1] => 1 => 1
[1,0,1,0,1,0]
 => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
 => [1,3,2] => 01 => 1
[1,1,0,0,1,0]
 => [2,1,3] => 10 => 1
[1,1,0,1,0,0]
 => [2,3,1] => 10 => 1
[1,1,1,0,0,0]
 => [3,1,2] => 10 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 010 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 010 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 100 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 100 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 100 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 100 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 110 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 100 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 100 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0010 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0010 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0100 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0100 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0100 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 0100 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 0100 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 0110 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 0100 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0100 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1010 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1010 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 1000 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 1001 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 1000 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 1000 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 1000 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 1000 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 1010 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 1000 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 1000 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 1000 => 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000010
(load all 2 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => [2] => [2]
 => 1 = 0 + 1
[1,1,0,0]
 => [2,1] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => [3] => [3]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1] => [2,1]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => [1,2] => [2,1]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => [2,1]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,2] => [2,1]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4] => [4]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1] => [3,1]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,2] => [2,2]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1] => [3,1]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,2] => [2,2]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3] => [3,1]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,2] => [2,2]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,1] => [3,1]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,2] => [2,2]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [1,3] => [3,1]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,2] => [2,2]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,3] => [3,1]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5] => [5]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,1] => [4,1]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1] => [4,1]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,3] => [3,2]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1] => [4,1]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [2,3] => [3,2]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,3] => [3,2]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,4] => [4,1]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,3,1] => [3,1,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,2,2] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,3,1] => [3,1,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [1,2,2] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3] => [3,2]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,1] => [4,1]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,3] => [3,2]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,3] => [3,2]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [1,4] => [4,1]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,3,5,6,8,4,7] => ? => ?
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,3,6,7,5,8] => ? => ?
 => ? = 3 + 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,1,4,6,7,3,5,8] => ? => ?
 => ? = 2 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [3,4,5,6,7,1,9,2,8] => ? => ?
 => ? = 2 + 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [3,8,1,2,4,5,9,6,7] => ? => ?
 => ? = 2 + 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4,8,1,2,3,9,5,6,7] => ? => ?
 => ? = 2 + 1
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,9,4,6,7,8] => ? => ?
 => ? = 2 + 1
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => [6,1,2,9,3,4,5,7,8] => ? => ?
 => ? = 2 + 1
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => [6,8,1,9,2,3,4,5,7] => ? => ?
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,2,5,6,7,3,8,9,4] => ? => ?
 => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,5,6,9,7,8] => ? => ?
 => ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,3,5,9,6,7,8] => ? => ?
 => ? = 2 + 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [5,6,7,8,9,11,1,2,3,4,10] => [6,5] => ?
 => ? = 1 + 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
 => [7,10,1,2,3,4,5,6,8,9] => ? => ?
 => ? = 1 + 1
[1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [6,10,1,2,3,4,5,7,8,9] => ? => ?
 => ? = 1 + 1
[1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [4,10,1,2,3,5,6,7,8,9] => ? => ?
 => ? = 1 + 1
[1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [2,12,1,3,4,5,6,7,8,9,10,11] => [2,10] => ?
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8,9,10,11] => ? => ?
 => ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => [3,4,5,6,7,8,9,10,1,2,11] => [8,3] => ?
 => ? = 1 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [6,7,8,9,10,1,2,3,4,5,11] => [5,6] => ?
 => ? = 1 + 1
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,7,8,9,10,11,2,3,4,5,6] => [6,5] => ?
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,11,1,7,8,9,10] => [6,5] => ?
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0]
 => [11,12,1,2,3,4,5,6,7,8,9,10] => [2,10] => ?
 => ? = 1 + 1
[1,1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [2,8,9,10,11,1,3,4,5,6,7] => [5,6] => ?
 => ? = 1 + 1

Description

The length of the partition.

Matching statistic: St000389
(load all 4 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => 0 => 0
[1,1,0,0]
 => [2,1] => 1 => 1
[1,0,1,0,1,0]
 => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
 => [1,3,2] => 01 => 1
[1,1,0,0,1,0]
 => [2,1,3] => 10 => 1
[1,1,0,1,0,0]
 => [2,3,1] => 01 => 1
[1,1,1,0,0,0]
 => [3,1,2] => 10 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 010 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 001 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 010 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 100 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 101 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 010 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 100 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0010 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0001 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0100 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0010 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 0010 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 0100 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 0101 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 0010 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0100 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1010 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0100 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0010 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0001 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0010 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 0100 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 0101 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 0010 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 0100 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 1000 => 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,2,4,6,3,7,8,5] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,3,8,5,7] => ? => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,2,5,3,7,8,4,6] => ? => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,2,5,6,8,3,4,7] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,2,6,7,3,4,8,5] => ? => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,4,7,2,5,8,6] => ? => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,3,5,2,6,4,8,7] => ? => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,3,6,7,2,4,8,5] => ? => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ? => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,3,5,6,8,4,7] => ? => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,3,6,7,5,8] => ? => ? = 3
[1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,1,4,6,7,3,5,8] => ? => ? = 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
 => [2,4,1,6,7,3,5,8] => ? => ? = 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [2,4,5,1,6,7,3,8] => ? => ? = 2
[1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [2,4,7,1,3,8,5,6] => ? => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [2,5,1,7,3,8,4,6] => ? => ? = 3
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [2,5,7,1,3,8,4,6] => ? => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [2,6,7,1,3,8,4,5] => ? => ? = 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,2,7,4,5,6,8] => ? => ? = 2
[1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
 => [3,4,5,1,7,2,6,8] => ? => ? = 2
[1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
 => [3,5,7,1,2,8,4,6] => ? => ? = 2
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,6,1,2,8,4,5,7] => ? => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
 => [4,1,2,6,3,5,7,8] => ? => ? = 2
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
 => [4,1,6,7,2,8,3,5] => ? => ? = 3
[1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
 => [4,1,7,2,3,8,5,6] => ? => ? = 3
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
 => [4,5,1,2,7,3,8,6] => ? => ? = 3
[1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
 => [4,6,1,7,2,3,8,5] => ? => ? = 3
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [5,1,2,3,7,4,6,8] => ? => ? = 2
[1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
 => [5,1,6,2,7,8,3,4] => ? => ? = 3
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [5,1,7,2,3,4,6,8] => ? => ? = 2
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
 => [5,6,1,7,2,8,3,4] => ? => ? = 3
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [2,3,6,7,1,8,9,4,5] => ? => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [3,4,5,6,7,1,9,2,8] => ? => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,6,7,9,1,2,8] => ? => ? = 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [3,8,1,2,4,5,9,6,7] => ? => ? = 2
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4,8,1,2,3,9,5,6,7] => ? => ? = 2
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,9,4,6,7,8] => ? => ? = 2
[1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,8,1,2,9,3,4,6,7] => ? => ? = 2
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => [6,1,2,9,3,4,5,7,8] => ? => ? = 2
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => [6,8,1,9,2,3,4,5,7] => ? => ? = 2
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [6,9,1,2,3,4,5,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,2,5,6,7,3,8,9,4] => ? => ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,5,6,9,7,8] => ? => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,3,5,9,6,7,8] => ? => ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
 => [10,1,2,3,4,5,6,7,8,11,9] => 1000000001 => ? = 2
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
 => [7,10,1,2,3,4,5,6,8,9] => ? => ? = 1
[1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [6,10,1,2,3,4,5,7,8,9] => ? => ? = 1
[1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [4,10,1,2,3,5,6,7,8,9] => ? => ? = 1
[1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [3,10,1,2,4,5,6,7,8,9] => ? => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0]
 => [11,1,2,3,4,5,6,7,8,9,12,10] => 10000000001 => ? = 2

Description

The number of runs of ones of odd length in a binary word.

Matching statistic: St000390
(load all 8 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => 0 => 0
[1,1,0,0]
 => [2,1] => 1 => 1
[1,0,1,0,1,0]
 => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
 => [1,3,2] => 01 => 1
[1,1,0,0,1,0]
 => [2,1,3] => 10 => 1
[1,1,0,1,0,0]
 => [2,3,1] => 01 => 1
[1,1,1,0,0,0]
 => [3,1,2] => 10 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 010 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 001 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 010 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 100 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 101 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 010 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 100 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0010 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0001 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0100 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0010 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 0010 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 0100 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 0101 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 0010 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0100 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1010 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0100 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0010 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0001 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0010 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 0100 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 0101 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 0010 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 0100 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 1000 => 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,2,4,6,3,7,8,5] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,3,8,5,7] => ? => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,2,5,3,7,8,4,6] => ? => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,2,5,6,8,3,4,7] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,2,6,7,3,4,8,5] => ? => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,4,7,2,5,8,6] => ? => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,3,5,2,6,4,8,7] => ? => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,3,6,7,2,4,8,5] => ? => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ? => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,3,5,6,8,4,7] => ? => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,3,6,7,5,8] => ? => ? = 3
[1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,1,4,6,7,3,5,8] => ? => ? = 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
 => [2,4,1,6,7,3,5,8] => ? => ? = 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [2,4,5,1,6,7,3,8] => ? => ? = 2
[1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [2,4,7,1,3,8,5,6] => ? => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [2,5,1,7,3,8,4,6] => ? => ? = 3
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [2,5,7,1,3,8,4,6] => ? => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [2,6,7,1,3,8,4,5] => ? => ? = 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,2,7,4,5,6,8] => ? => ? = 2
[1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
 => [3,4,5,1,7,2,6,8] => ? => ? = 2
[1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
 => [3,5,7,1,2,8,4,6] => ? => ? = 2
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,6,1,2,8,4,5,7] => ? => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
 => [4,1,2,6,3,5,7,8] => ? => ? = 2
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
 => [4,1,6,7,2,8,3,5] => ? => ? = 3
[1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
 => [4,1,7,2,3,8,5,6] => ? => ? = 3
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
 => [4,5,1,2,7,3,8,6] => ? => ? = 3
[1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
 => [4,6,1,7,2,3,8,5] => ? => ? = 3
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [5,1,2,3,7,4,6,8] => ? => ? = 2
[1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
 => [5,1,6,2,7,8,3,4] => ? => ? = 3
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [5,1,7,2,3,4,6,8] => ? => ? = 2
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
 => [5,6,1,7,2,8,3,4] => ? => ? = 3
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [2,3,6,7,1,8,9,4,5] => ? => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [3,4,5,6,7,1,9,2,8] => ? => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,6,7,9,1,2,8] => ? => ? = 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [3,8,1,2,4,5,9,6,7] => ? => ? = 2
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4,8,1,2,3,9,5,6,7] => ? => ? = 2
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,9,4,6,7,8] => ? => ? = 2
[1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,8,1,2,9,3,4,6,7] => ? => ? = 2
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => [6,1,2,9,3,4,5,7,8] => ? => ? = 2
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => [6,8,1,9,2,3,4,5,7] => ? => ? = 2
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [6,9,1,2,3,4,5,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,2,5,6,7,3,8,9,4] => ? => ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,5,6,9,7,8] => ? => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,3,5,9,6,7,8] => ? => ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
 => [10,1,2,3,4,5,6,7,8,11,9] => 1000000001 => ? = 2
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
 => [7,10,1,2,3,4,5,6,8,9] => ? => ? = 1
[1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [6,10,1,2,3,4,5,7,8,9] => ? => ? = 1
[1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [4,10,1,2,3,5,6,7,8,9] => ? => ? = 1
[1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [3,10,1,2,4,5,6,7,8,9] => ? => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0]
 => [11,1,2,3,4,5,6,7,8,9,12,10] => 10000000001 => ? = 2

Description

The number of runs of ones in a binary word.

Matching statistic: St000566

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => [1,1]
 => 0
[1,1,0,0]
 => [2,1] => [2]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [2,1,1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,2,4,3,7,5,8,6] => ?
 => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,3,8,5,7] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,2,5,3,7,8,4,6] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,2,5,6,3,4,8,7] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,2,5,6,8,3,4,7] => ?
 => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,2,6,7,3,4,8,5] => ?
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,3,4,6,7,8,2,5] => ?
 => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,4,7,2,5,8,6] => ?
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,3,5,2,6,4,8,7] => ?
 => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,3,6,7,2,4,8,5] => ?
 => ? = 2
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,4,5,8,2,3,6,7] => ?
 => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,4,6,7,2,3,8,5] => ?
 => ? = 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,6,7,2,3,4,8,5] => ?
 => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ?
 => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,3,5,6,8,4,7] => ?
 => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,3,6,7,5,8] => ?
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [2,3,4,6,1,7,5,8] => ?
 => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [2,3,5,1,6,4,7,8] => ?
 => ? = 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [2,4,1,3,5,7,8,6] => ?
 => ? = 2
[1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [2,4,5,8,1,3,6,7] => ?
 => ? = 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [2,5,1,3,7,4,6,8] => ?
 => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [2,6,7,1,3,8,4,5] => ?
 => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,7,1,3,4,8,5,6] => ?
 => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,6,2,7,8] => ?
 => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
 => [3,1,6,2,4,7,5,8] => ?
 => ? = 3
[1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2,8,6,7] => ?
 => ? = 2
[1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
 => [3,5,1,7,2,4,8,6] => ?
 => ? = 3
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,6,1,2,8,4,5,7] => ?
 => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
 => [4,1,2,6,3,5,7,8] => ?
 => ? = 2
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
 => [4,1,5,8,2,3,6,7] => ?
 => ? = 2
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
 => [4,1,6,7,2,8,3,5] => ?
 => ? = 3
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
 => [4,5,1,2,7,3,8,6] => ?
 => ? = 3
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [5,1,2,3,7,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
 => [5,1,2,6,3,4,8,7] => ?
 => ? = 3
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
 => [5,1,2,7,3,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [5,1,7,2,3,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
 => [5,1,7,2,3,8,4,6] => ?
 => ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
 => [5,7,1,2,3,8,4,6] => ?
 => ? = 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [5,7,1,8,2,3,4,6] => ?
 => ? = 2
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [2,3,6,1,7,8,9,4,5] => ?
 => ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [2,3,6,7,1,8,9,4,5] => ?
 => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [2,4,5,6,7,8,1,9,3] => ?
 => ? = 2
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [2,6,1,3,7,8,9,4,5] => ?
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [3,4,5,6,7,1,9,2,8] => ?
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,6,7,9,1,2,8] => ?
 => ? = 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [3,8,1,2,4,5,9,6,7] => ?
 => ? = 2
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4,8,1,2,3,9,5,6,7] => ?
 => ? = 2
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,9,4,6,7,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
 => [5,1,7,2,9,3,4,6,8] => ?
 => ? = 3
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [5,6,1,7,2,8,3,9,4] => ?
 => ? = 4

Description

The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if

$\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is

$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$

Matching statistic: St001251

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => [1,1]
 => 0
[1,1,0,0]
 => [2,1] => [2]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [2,1,1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,2,4,3,7,5,8,6] => ?
 => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,3,8,5,7] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,2,5,3,7,8,4,6] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,2,5,6,3,4,8,7] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,2,5,6,8,3,4,7] => ?
 => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,2,6,7,3,4,8,5] => ?
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,3,4,6,7,8,2,5] => ?
 => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,4,7,2,5,8,6] => ?
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,3,5,2,6,4,8,7] => ?
 => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,3,6,7,2,4,8,5] => ?
 => ? = 2
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,4,5,8,2,3,6,7] => ?
 => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,4,6,7,2,3,8,5] => ?
 => ? = 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,6,7,2,3,4,8,5] => ?
 => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ?
 => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,3,5,6,8,4,7] => ?
 => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,3,6,7,5,8] => ?
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [2,3,4,6,1,7,5,8] => ?
 => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [2,3,5,1,6,4,7,8] => ?
 => ? = 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [2,4,1,3,5,7,8,6] => ?
 => ? = 2
[1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [2,4,5,8,1,3,6,7] => ?
 => ? = 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [2,5,1,3,7,4,6,8] => ?
 => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [2,6,7,1,3,8,4,5] => ?
 => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,7,1,3,4,8,5,6] => ?
 => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,6,2,7,8] => ?
 => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
 => [3,1,6,2,4,7,5,8] => ?
 => ? = 3
[1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2,8,6,7] => ?
 => ? = 2
[1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
 => [3,5,1,7,2,4,8,6] => ?
 => ? = 3
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,6,1,2,8,4,5,7] => ?
 => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
 => [4,1,2,6,3,5,7,8] => ?
 => ? = 2
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
 => [4,1,5,8,2,3,6,7] => ?
 => ? = 2
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
 => [4,1,6,7,2,8,3,5] => ?
 => ? = 3
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
 => [4,5,1,2,7,3,8,6] => ?
 => ? = 3
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [5,1,2,3,7,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
 => [5,1,2,6,3,4,8,7] => ?
 => ? = 3
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
 => [5,1,2,7,3,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [5,1,7,2,3,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
 => [5,1,7,2,3,8,4,6] => ?
 => ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
 => [5,7,1,2,3,8,4,6] => ?
 => ? = 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [5,7,1,8,2,3,4,6] => ?
 => ? = 2
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [2,3,6,1,7,8,9,4,5] => ?
 => ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [2,3,6,7,1,8,9,4,5] => ?
 => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [2,4,5,6,7,8,1,9,3] => ?
 => ? = 2
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [2,6,1,3,7,8,9,4,5] => ?
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [3,4,5,6,7,1,9,2,8] => ?
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,6,7,9,1,2,8] => ?
 => ? = 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [3,8,1,2,4,5,9,6,7] => ?
 => ? = 2
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4,8,1,2,3,9,5,6,7] => ?
 => ? = 2
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,9,4,6,7,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
 => [5,1,7,2,9,3,4,6,8] => ?
 => ? = 3
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [5,6,1,7,2,8,3,9,4] => ?
 => ? = 4

Description

The number of parts of a partition that are not congruent 1 modulo 3.

Matching statistic: St001252

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => [1,1]
 => 0
[1,1,0,0]
 => [2,1] => [2]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [2,1,1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,2,4,3,7,5,8,6] => ?
 => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,3,8,5,7] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,2,5,3,7,8,4,6] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,2,5,6,3,4,8,7] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,2,5,6,8,3,4,7] => ?
 => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,2,6,7,3,4,8,5] => ?
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,3,4,6,7,8,2,5] => ?
 => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,4,7,2,5,8,6] => ?
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,3,5,2,6,4,8,7] => ?
 => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,3,6,7,2,4,8,5] => ?
 => ? = 2
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,4,5,8,2,3,6,7] => ?
 => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,4,6,7,2,3,8,5] => ?
 => ? = 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,6,7,2,3,4,8,5] => ?
 => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ?
 => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,3,5,6,8,4,7] => ?
 => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,3,6,7,5,8] => ?
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [2,3,4,6,1,7,5,8] => ?
 => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [2,3,5,1,6,4,7,8] => ?
 => ? = 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [2,4,1,3,5,7,8,6] => ?
 => ? = 2
[1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [2,4,5,8,1,3,6,7] => ?
 => ? = 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [2,5,1,3,7,4,6,8] => ?
 => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [2,6,7,1,3,8,4,5] => ?
 => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,7,1,3,4,8,5,6] => ?
 => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,6,2,7,8] => ?
 => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
 => [3,1,6,2,4,7,5,8] => ?
 => ? = 3
[1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2,8,6,7] => ?
 => ? = 2
[1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
 => [3,5,1,7,2,4,8,6] => ?
 => ? = 3
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,6,1,2,8,4,5,7] => ?
 => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
 => [4,1,2,6,3,5,7,8] => ?
 => ? = 2
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
 => [4,1,5,8,2,3,6,7] => ?
 => ? = 2
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
 => [4,1,6,7,2,8,3,5] => ?
 => ? = 3
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
 => [4,5,1,2,7,3,8,6] => ?
 => ? = 3
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [5,1,2,3,7,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
 => [5,1,2,6,3,4,8,7] => ?
 => ? = 3
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
 => [5,1,2,7,3,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [5,1,7,2,3,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
 => [5,1,7,2,3,8,4,6] => ?
 => ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
 => [5,7,1,2,3,8,4,6] => ?
 => ? = 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [5,7,1,8,2,3,4,6] => ?
 => ? = 2
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [2,3,6,1,7,8,9,4,5] => ?
 => ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [2,3,6,7,1,8,9,4,5] => ?
 => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [2,4,5,6,7,8,1,9,3] => ?
 => ? = 2
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [2,6,1,3,7,8,9,4,5] => ?
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [3,4,5,6,7,1,9,2,8] => ?
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,6,7,9,1,2,8] => ?
 => ? = 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [3,8,1,2,4,5,9,6,7] => ?
 => ? = 2
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4,8,1,2,3,9,5,6,7] => ?
 => ? = 2
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,9,4,6,7,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
 => [5,1,7,2,9,3,4,6,8] => ?
 => ? = 3
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [5,6,1,7,2,8,3,9,4] => ?
 => ? = 4

Description

Half the sum of the even parts of a partition.

Matching statistic: St001280
(load all 5 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => [1,1]
 => 0
[1,1,0,0]
 => [2,1] => [2]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [2,1,1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,2,4,3,7,5,8,6] => ?
 => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,3,8,5,7] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,2,5,3,7,8,4,6] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,2,5,6,3,4,8,7] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,2,5,6,8,3,4,7] => ?
 => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,2,6,7,3,4,8,5] => ?
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,3,4,6,7,8,2,5] => ?
 => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,4,7,2,5,8,6] => ?
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,3,5,2,6,4,8,7] => ?
 => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,3,6,7,2,4,8,5] => ?
 => ? = 2
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,4,5,8,2,3,6,7] => ?
 => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,4,6,7,2,3,8,5] => ?
 => ? = 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,6,7,2,3,4,8,5] => ?
 => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ?
 => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,3,5,6,8,4,7] => ?
 => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,3,6,7,5,8] => ?
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [2,3,4,6,1,7,5,8] => ?
 => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [2,3,5,1,6,4,7,8] => ?
 => ? = 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [2,4,1,3,5,7,8,6] => ?
 => ? = 2
[1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [2,4,5,8,1,3,6,7] => ?
 => ? = 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [2,5,1,3,7,4,6,8] => ?
 => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [2,6,7,1,3,8,4,5] => ?
 => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,7,1,3,4,8,5,6] => ?
 => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,6,2,7,8] => ?
 => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
 => [3,1,6,2,4,7,5,8] => ?
 => ? = 3
[1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2,8,6,7] => ?
 => ? = 2
[1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
 => [3,5,1,7,2,4,8,6] => ?
 => ? = 3
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,6,1,2,8,4,5,7] => ?
 => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
 => [4,1,2,6,3,5,7,8] => ?
 => ? = 2
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
 => [4,1,5,8,2,3,6,7] => ?
 => ? = 2
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
 => [4,1,6,7,2,8,3,5] => ?
 => ? = 3
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
 => [4,5,1,2,7,3,8,6] => ?
 => ? = 3
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [5,1,2,3,7,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
 => [5,1,2,6,3,4,8,7] => ?
 => ? = 3
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
 => [5,1,2,7,3,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [5,1,7,2,3,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
 => [5,1,7,2,3,8,4,6] => ?
 => ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
 => [5,7,1,2,3,8,4,6] => ?
 => ? = 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [5,7,1,8,2,3,4,6] => ?
 => ? = 2
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [2,3,6,1,7,8,9,4,5] => ?
 => ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [2,3,6,7,1,8,9,4,5] => ?
 => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [2,4,5,6,7,8,1,9,3] => ?
 => ? = 2
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [2,6,1,3,7,8,9,4,5] => ?
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [3,4,5,6,7,1,9,2,8] => ?
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,6,7,9,1,2,8] => ?
 => ? = 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [3,8,1,2,4,5,9,6,7] => ?
 => ? = 2
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4,8,1,2,3,9,5,6,7] => ?
 => ? = 2
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,9,4,6,7,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
 => [5,1,7,2,9,3,4,6,8] => ?
 => ? = 3
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [5,6,1,7,2,8,3,9,4] => ?
 => ? = 4

Description

The number of parts of an integer partition that are at least two.

Matching statistic: St001657

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => [1,1]
 => 0
[1,1,0,0]
 => [2,1] => [2]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [2,1,1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,2,4,3,7,5,8,6] => ?
 => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,3,8,5,7] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,2,5,3,7,8,4,6] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,2,5,6,3,4,8,7] => ?
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,2,5,6,8,3,4,7] => ?
 => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,2,6,7,3,4,8,5] => ?
 => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,3,4,6,7,8,2,5] => ?
 => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,4,7,2,5,8,6] => ?
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,3,5,2,6,4,8,7] => ?
 => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,3,6,7,2,4,8,5] => ?
 => ? = 2
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,4,5,8,2,3,6,7] => ?
 => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,4,6,7,2,3,8,5] => ?
 => ? = 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,6,7,2,3,4,8,5] => ?
 => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ?
 => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,3,5,6,8,4,7] => ?
 => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,3,6,7,5,8] => ?
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [2,3,4,6,1,7,5,8] => ?
 => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [2,3,5,1,6,4,7,8] => ?
 => ? = 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [2,4,1,3,5,7,8,6] => ?
 => ? = 2
[1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [2,4,5,8,1,3,6,7] => ?
 => ? = 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [2,5,1,3,7,4,6,8] => ?
 => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [2,6,7,1,3,8,4,5] => ?
 => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,7,1,3,4,8,5,6] => ?
 => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,6,2,7,8] => ?
 => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
 => [3,1,6,2,4,7,5,8] => ?
 => ? = 3
[1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2,8,6,7] => ?
 => ? = 2
[1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
 => [3,5,1,7,2,4,8,6] => ?
 => ? = 3
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,6,1,2,8,4,5,7] => ?
 => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
 => [4,1,2,6,3,5,7,8] => ?
 => ? = 2
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
 => [4,1,5,8,2,3,6,7] => ?
 => ? = 2
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
 => [4,1,6,7,2,8,3,5] => ?
 => ? = 3
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
 => [4,5,1,2,7,3,8,6] => ?
 => ? = 3
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [5,1,2,3,7,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
 => [5,1,2,6,3,4,8,7] => ?
 => ? = 3
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
 => [5,1,2,7,3,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [5,1,7,2,3,4,6,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
 => [5,1,7,2,3,8,4,6] => ?
 => ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
 => [5,7,1,2,3,8,4,6] => ?
 => ? = 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [5,7,1,8,2,3,4,6] => ?
 => ? = 2
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [2,3,6,1,7,8,9,4,5] => ?
 => ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [2,3,6,7,1,8,9,4,5] => ?
 => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [2,4,5,6,7,8,1,9,3] => ?
 => ? = 2
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [2,6,1,3,7,8,9,4,5] => ?
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [3,4,5,6,7,1,9,2,8] => ?
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,6,7,9,1,2,8] => ?
 => ? = 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [3,8,1,2,4,5,9,6,7] => ?
 => ? = 2
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4,8,1,2,3,9,5,6,7] => ?
 => ? = 2
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,9,4,6,7,8] => ?
 => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
 => [5,1,7,2,9,3,4,6,8] => ?
 => ? = 3
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [5,6,1,7,2,8,3,9,4] => ?
 => ? = 4

Description

The number of twos in an integer partition. The total number of twos in all partitions of

$n$ is equal to the total number of singletons [[St001484]] in all partitions of

$n-1$ , see [1].

Matching statistic: St000345

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => [1,1]
 => 1 = 0 + 1
[1,1,0,0]
 => [2,1] => [2]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,1,1]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,2,4,3,7,5,8,6] => ?
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,3,8,5,7] => ?
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,2,5,3,7,8,4,6] => ?
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,2,5,6,3,4,8,7] => ?
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,2,5,6,8,3,4,7] => ?
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,2,6,7,3,4,8,5] => ?
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,3,4,6,7,8,2,5] => ?
 => ? = 1 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,4,7,2,5,8,6] => ?
 => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,3,5,2,6,4,8,7] => ?
 => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,3,6,7,2,4,8,5] => ?
 => ? = 2 + 1
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,4,5,8,2,3,6,7] => ?
 => ? = 1 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,4,6,7,2,3,8,5] => ?
 => ? = 2 + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,6,7,2,3,4,8,5] => ?
 => ? = 2 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,5,4,6,7,8] => ?
 => ? = 2 + 1
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,3,5,6,8,4,7] => ?
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,3,6,7,5,8] => ?
 => ? = 3 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [2,3,4,6,1,7,5,8] => ?
 => ? = 2 + 1
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [2,3,5,1,6,4,7,8] => ?
 => ? = 2 + 1
[1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [2,4,1,3,5,7,8,6] => ?
 => ? = 2 + 1
[1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [2,4,5,8,1,3,6,7] => ?
 => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [2,5,1,3,7,4,6,8] => ?
 => ? = 2 + 1
[1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [2,6,7,1,3,8,4,5] => ?
 => ? = 2 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [2,7,1,3,4,8,5,6] => ?
 => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,6,2,7,8] => ?
 => ? = 2 + 1
[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
 => [3,1,6,2,4,7,5,8] => ?
 => ? = 3 + 1
[1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2,8,6,7] => ?
 => ? = 2 + 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
 => [3,5,1,7,2,4,8,6] => ?
 => ? = 3 + 1
[1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => [3,6,1,2,8,4,5,7] => ?
 => ? = 2 + 1
[1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
 => [4,1,2,6,3,5,7,8] => ?
 => ? = 2 + 1
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
 => [4,1,5,8,2,3,6,7] => ?
 => ? = 2 + 1
[1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
 => [4,1,6,7,2,8,3,5] => ?
 => ? = 3 + 1
[1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
 => [4,5,1,2,7,3,8,6] => ?
 => ? = 3 + 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [5,1,2,3,7,4,6,8] => ?
 => ? = 2 + 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
 => [5,1,2,6,3,4,8,7] => ?
 => ? = 3 + 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
 => [5,1,2,7,3,4,6,8] => ?
 => ? = 2 + 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [5,1,7,2,3,4,6,8] => ?
 => ? = 2 + 1
[1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
 => [5,1,7,2,3,8,4,6] => ?
 => ? = 3 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
 => [5,7,1,2,3,8,4,6] => ?
 => ? = 2 + 1
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [5,7,1,8,2,3,4,6] => ?
 => ? = 2 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [2,3,6,1,7,8,9,4,5] => ?
 => ? = 2 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [2,3,6,7,1,8,9,4,5] => ?
 => ? = 2 + 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [2,4,5,6,7,8,1,9,3] => ?
 => ? = 2 + 1
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [2,6,1,3,7,8,9,4,5] => ?
 => ? = 2 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [3,4,5,6,7,1,9,2,8] => ?
 => ? = 2 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,6,7,9,1,2,8] => ?
 => ? = 1 + 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [3,8,1,2,4,5,9,6,7] => ?
 => ? = 2 + 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [4,8,1,2,3,9,5,6,7] => ?
 => ? = 2 + 1
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,9,4,6,7,8] => ?
 => ? = 2 + 1
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
 => [5,1,7,2,9,3,4,6,8] => ?
 => ? = 3 + 1
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [5,6,1,7,2,8,3,9,4] => ?
 => ? = 4 + 1

Description

The number of refinements of a partition. A partition

$\lambda$ refines a partition

$\mu$ if the parts of

$\mu$ can be subdivided to obtain the parts of

$\lambda$ .

The following 168 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000377The dinv defect of an integer partition. St001176The size of a partition minus its first part. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000877The depth of the binary word interpreted as a path. St000326The position of the first one in a binary word after appending a 1 at the end. St000150The floored half-sum of the multiplicities of a partition. St000142The number of even parts of a partition. St000185The weighted size of a partition. St000507The number of ascents of a standard tableau. St000321The number of integer partitions of n that are dominated by an integer partition. St000291The number of descents of a binary word. St000157The number of descents of a standard tableau. St000097The order of the largest clique of the graph. St001581The achromatic number of a graph. St000011The number of touch points (or returns) of a Dyck path. St000919The number of maximal left branches of a binary tree. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000306The bounce count of a Dyck path. St000336The leg major index of a standard tableau. St000098The chromatic number of a graph. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000251The number of nonsingleton blocks of a set partition. St000245The number of ascents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000211The rank of the set partition. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000172The Grundy number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001471The magnitude of a Dyck path. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001512The minimum rank of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St000665The number of rafts of a permutation. St000834The number of right outer peaks of a permutation. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000374The number of exclusive right-to-left minima of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000167The number of leaves of an ordered tree. St000670The reversal length of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000035The number of left outer peaks of a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St001726The number of visible inversions of a permutation. St000809The reduced reflection length of the permutation. St000662The staircase size of the code of a permutation. St000386The number of factors DDU in a Dyck path. St000742The number of big ascents of a permutation after prepending zero. St000141The maximum drop size of a permutation. St000632The jump number of the poset. St000201The number of leaf nodes in a binary tree. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000702The number of weak deficiencies of a permutation. St001427The number of descents of a signed permutation. St000647The number of big descents of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St000083The number of left oriented leafs of a binary tree except the first one. St000168The number of internal nodes of an ordered tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001812The biclique partition number of a graph. St001840The number of descents of a set partition. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000619The number of cyclic descents of a permutation. St000353The number of inner valleys of a permutation. St001330The hat guessing number of a graph. St001469The holeyness of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000640The rank of the largest boolean interval in a poset. St000256The number of parts from which one can substract 2 and still get an integer partition. St000646The number of big ascents of a permutation. St000092The number of outer peaks of a permutation. St000023The number of inner peaks of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000099The number of valleys of a permutation, including the boundary. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000455The second largest eigenvalue of a graph if it is integral. St000534The number of 2-rises of a permutation. St001597The Frobenius rank of a skew partition. St001946The number of descents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one.