- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000150
(load all 3 compositions to match this statistic)

Mapping

St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].

Matching statistic: St000288

Mapping

Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%

Values

[1]
 => [1]
 => 1 => 0 => 0
[2]
 => [1,1]
 => 11 => 00 => 0
[1,1]
 => [2]
 => 0 => 1 => 1
[3]
 => [3]
 => 1 => 0 => 0
[2,1]
 => [1,1,1]
 => 111 => 000 => 0
[1,1,1]
 => [2,1]
 => 01 => 10 => 1
[4]
 => [1,1,1,1]
 => 1111 => 0000 => 0
[3,1]
 => [3,1]
 => 11 => 00 => 0
[2,2]
 => [4]
 => 0 => 1 => 1
[2,1,1]
 => [2,1,1]
 => 011 => 100 => 1
[1,1,1,1]
 => [2,2]
 => 00 => 11 => 2
[5]
 => [5]
 => 1 => 0 => 0
[4,1]
 => [1,1,1,1,1]
 => 11111 => 00000 => 0
[3,2]
 => [3,1,1]
 => 111 => 000 => 0
[3,1,1]
 => [3,2]
 => 10 => 01 => 1
[2,2,1]
 => [4,1]
 => 01 => 10 => 1
[2,1,1,1]
 => [2,1,1,1]
 => 0111 => 1000 => 1
[1,1,1,1,1]
 => [2,2,1]
 => 001 => 110 => 2
[6]
 => [3,3]
 => 11 => 00 => 0
[5,1]
 => [5,1]
 => 11 => 00 => 0
[4,2]
 => [1,1,1,1,1,1]
 => 111111 => 000000 => 0
[4,1,1]
 => [2,1,1,1,1]
 => 01111 => 10000 => 1
[3,3]
 => [6]
 => 0 => 1 => 1
[3,2,1]
 => [3,1,1,1]
 => 1111 => 0000 => 0
[3,1,1,1]
 => [3,2,1]
 => 101 => 010 => 1
[2,2,2]
 => [4,1,1]
 => 011 => 100 => 1
[2,2,1,1]
 => [4,2]
 => 00 => 11 => 2
[2,1,1,1,1]
 => [2,2,1,1]
 => 0011 => 1100 => 2
[1,1,1,1,1,1]
 => [2,2,2]
 => 000 => 111 => 3
[7]
 => [7]
 => 1 => 0 => 0
[6,1]
 => [3,3,1]
 => 111 => 000 => 0
[5,2]
 => [5,1,1]
 => 111 => 000 => 0
[5,1,1]
 => [5,2]
 => 10 => 01 => 1
[4,3]
 => [3,1,1,1,1]
 => 11111 => 00000 => 0
[4,2,1]
 => [1,1,1,1,1,1,1]
 => 1111111 => 0000000 => 0
[4,1,1,1]
 => [2,1,1,1,1,1]
 => 011111 => 100000 => 1
[3,3,1]
 => [6,1]
 => 01 => 10 => 1
[3,2,2]
 => [4,3]
 => 01 => 10 => 1
[3,2,1,1]
 => [3,2,1,1]
 => 1011 => 0100 => 1
[3,1,1,1,1]
 => [3,2,2]
 => 100 => 011 => 2
[2,2,2,1]
 => [4,1,1,1]
 => 0111 => 1000 => 1
[2,2,1,1,1]
 => [4,2,1]
 => 001 => 110 => 2
[2,1,1,1,1,1]
 => [2,2,1,1,1]
 => 00111 => 11000 => 2
[1,1,1,1,1,1,1]
 => [2,2,2,1]
 => 0001 => 1110 => 3
[8]
 => [1,1,1,1,1,1,1,1]
 => 11111111 => 00000000 => 0
[7,1]
 => [7,1]
 => 11 => 00 => 0
[6,2]
 => [3,3,1,1]
 => 1111 => 0000 => 0
[6,1,1]
 => [3,3,2]
 => 110 => 001 => 1
[5,3]
 => [5,3]
 => 11 => 00 => 0
[5,2,1]
 => [5,1,1,1]
 => 1111 => 0000 => 0
[8,2]
 => [1,1,1,1,1,1,1,1,1,1]
 => 1111111111 => 0000000000 => ? = 0
[8,2,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => 11111111111 => 00000000000 => ? = 0
[8,4]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => 111111111111 => 000000000000 => ? = 0
[8,3,1]
 => [3,1,1,1,1,1,1,1,1,1]
 => 1111111111 => 0000000000 => ? = 0
[8,1,1,1,1]
 => [2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => 1100000000 => ? = 2
[8,4,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => 1111111111111 => ? => ? = 0
[8,3,2]
 => [3,1,1,1,1,1,1,1,1,1,1]
 => 11111111111 => 00000000000 => ? = 0
[8,6]
 => [3,3,1,1,1,1,1,1,1,1]
 => 1111111111 => 0000000000 => ? = 0
[8,5,1]
 => [5,1,1,1,1,1,1,1,1,1]
 => 1111111111 => 0000000000 => ? = 0
[8,4,2]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => 11111111111111 => ? => ? = 0
[8,1,1,1,1,1,1]
 => [2,2,2,1,1,1,1,1,1,1,1]
 => 00011111111 => 11100000000 => ? = 3
[8,5,2]
 => [5,1,1,1,1,1,1,1,1,1,1]
 => 11111111111 => 00000000000 => ? = 0
[16]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => 1111111111111111 => ? => ? = 0
[8,6,2]
 => [3,3,1,1,1,1,1,1,1,1,1,1]
 => 111111111111 => 000000000000 => ? = 0
[8,5,3]
 => [5,3,1,1,1,1,1,1,1,1]
 => 1111111111 => 0000000000 => ? = 0
[16,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => 11111111111111111 => ? => ? = 0
[8,6,3]
 => [3,3,3,1,1,1,1,1,1,1,1]
 => 11111111111 => 00000000000 => ? = 0
[]
 => ?
 => ? => ? => ? = 0
[6,5,4,3,2,1]
 => ?
 => ? => ? => ? = 0
[5,5,4,3,2,1]
 => ?
 => ? => ? => ? = 1
[6,4,4,3,2,1]
 => ?
 => ? => ? => ? = 1
[5,4,4,3,2,1]
 => ?
 => ? => ? => ? = 1
[4,4,4,3,2,1]
 => ?
 => ? => ? => ? = 1
[6,5,3,3,2,1]
 => ?
 => ? => ? => ? = 1
[5,5,3,3,2,1]
 => ?
 => ? => ? => ? = 2
[5,4,3,3,2,1]
 => ?
 => ? => ? => ? = 1
[6,3,3,3,2,1]
 => ?
 => ? => ? => ? = 1
[6,5,4,2,2,1]
 => ?
 => ? => ? => ? = 1
[5,5,4,2,2,1]
 => ?
 => ? => ? => ? = 2
[6,4,4,2,2,1]
 => ?
 => ? => ? => ? = 2
[6,5,2,2,2,1]
 => ?
 => ? => ? => ? = 1
[6,5,4,3,1,1]
 => ?
 => ? => ? => ? = 1
[5,5,4,3,1,1]
 => ?
 => ? => ? => ? = 2
[6,4,4,3,1,1]
 => ?
 => ? => ? => ? = 2
[6,5,3,3,1,1]
 => ?
 => ? => ? => ? = 2
[5,5,3,3,1,1]
 => ?
 => ? => ? => ? = 3
[6,5,4,1,1,1]
 => ?
 => ? => ? => ? = 1
[6,5,4,3,2]
 => ?
 => ? => ? => ? = 0
[5,5,4,3,2]
 => ?
 => ? => ? => ? = 1
[6,4,4,3,2]
 => ?
 => ? => ? => ? = 1
[6,5,3,3,2]
 => ?
 => ? => ? => ? = 1
[5,5,3,3,2]
 => ?
 => ? => ? => ? = 2
[6,5,4,2,2]
 => ?
 => ? => ? => ? = 1
[5,5,4,2,2]
 => ?
 => ? => ? => ? = 2
[6,4,4,2,2]
 => ?
 => ? => ? => ? = 2
[6,5,4,3,1]
 => ?
 => ? => ? => ? = 0
[6,5,4,3]
 => ?
 => ? => ? => ? = 0
[7,6,5,4,3,2,1]
 => ?
 => ? => ? => ? = 0
[6,6,5,4,3,2,1]
 => ?
 => ? => ? => ? = 1
[6,5,5,4,3,2,1]
 => ?
 => ? => ? => ? = 1

Description

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

Matching statistic: St000992

Mapping

Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%

Values

[1]
 => [1]
 => [1]
 => []
 => 0
[2]
 => [1,1]
 => [2]
 => []
 => 0
[1,1]
 => [2]
 => [1,1]
 => [1]
 => 1
[3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 0
[2,1]
 => [1,1,1]
 => [3]
 => []
 => 0
[1,1,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[4]
 => [1,1,1,1]
 => [4]
 => []
 => 0
[3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,2]
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
[1,1,1,1]
 => [2,2]
 => [2,2]
 => [2]
 => 2
[5]
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[4,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => 0
[3,2]
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[3,1,1]
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 1
[2,2,1]
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[1,1,1,1,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 2
[6]
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 0
[5,1]
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[4,2]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => 0
[4,1,1]
 => [2,1,1,1,1]
 => [5,1]
 => [1]
 => 1
[3,3]
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[3,2,1]
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 0
[3,1,1,1]
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 1
[2,2,2]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
[2,2,1,1]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 2
[2,1,1,1,1]
 => [2,2,1,1]
 => [4,2]
 => [2]
 => 2
[1,1,1,1,1,1]
 => [2,2,2]
 => [3,3]
 => [3]
 => 3
[7]
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0
[6,1]
 => [3,3,1]
 => [3,2,2]
 => [2,2]
 => 0
[5,2]
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 0
[5,1,1]
 => [5,2]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 1
[4,3]
 => [3,1,1,1,1]
 => [5,1,1]
 => [1,1]
 => 0
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [7]
 => []
 => 0
[4,1,1,1]
 => [2,1,1,1,1,1]
 => [6,1]
 => [1]
 => 1
[3,3,1]
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[3,2,2]
 => [4,3]
 => [2,2,2,1]
 => [2,2,1]
 => 1
[3,2,1,1]
 => [3,2,1,1]
 => [4,2,1]
 => [2,1]
 => 1
[3,1,1,1,1]
 => [3,2,2]
 => [3,3,1]
 => [3,1]
 => 2
[2,2,2,1]
 => [4,1,1,1]
 => [4,1,1,1]
 => [1,1,1]
 => 1
[2,2,1,1,1]
 => [4,2,1]
 => [3,2,1,1]
 => [2,1,1]
 => 2
[2,1,1,1,1,1]
 => [2,2,1,1,1]
 => [5,2]
 => [2]
 => 2
[1,1,1,1,1,1,1]
 => [2,2,2,1]
 => [4,3]
 => [3]
 => 3
[8]
 => [1,1,1,1,1,1,1,1]
 => [8]
 => []
 => 0
[7,1]
 => [7,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0
[6,2]
 => [3,3,1,1]
 => [4,2,2]
 => [2,2]
 => 0
[6,1,1]
 => [3,3,2]
 => [3,3,2]
 => [3,2]
 => 1
[5,3]
 => [5,3]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => 0
[5,2,1]
 => [5,1,1,1]
 => [4,1,1,1,1]
 => [1,1,1,1]
 => 0
[7,7]
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
[15]
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[11,2,2]
 => [11,4]
 => [2,2,2,2,1,1,1,1,1,1,1]
 => [2,2,2,1,1,1,1,1,1,1]
 => ? = 1
[6,6,3]
 => [12,3]
 => [2,2,2,1,1,1,1,1,1,1,1,1]
 => [2,2,1,1,1,1,1,1,1,1,1]
 => ? = 1
[5,5,5]
 => [10,5]
 => [2,2,2,2,2,1,1,1,1,1]
 => [2,2,2,2,1,1,1,1,1]
 => ? = 1
[15,1]
 => [15,1]
 => [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[8,8]
 => [16]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
[6,6,2,2]
 => [12,4]
 => [2,2,2,2,1,1,1,1,1,1,1,1]
 => [2,2,2,1,1,1,1,1,1,1,1]
 => ? = 2
[6,5,5]
 => [10,3,3]
 => [3,3,3,1,1,1,1,1,1,1]
 => [3,3,1,1,1,1,1,1,1]
 => ? = 1
[5,5,3,3]
 => [10,6]
 => [2,2,2,2,2,2,1,1,1,1]
 => [2,2,2,2,2,1,1,1,1]
 => ? = 2
[5,4,4,3]
 => [8,5,3]
 => [3,3,3,2,2,1,1,1]
 => [3,3,2,2,1,1,1]
 => ? = 1
[4,4,4,4]
 => [8,8]
 => [2,2,2,2,2,2,2,2]
 => [2,2,2,2,2,2,2]
 => ? = 2
[4,4,2,2,2,2]
 => [8,4,4]
 => [3,3,3,3,1,1,1,1]
 => [3,3,3,1,1,1,1]
 => ? = 3
[3,3,3,3,2,2]
 => [6,6,4]
 => [3,3,3,3,2,2]
 => [3,3,3,2,2]
 => ? = 3
[17]
 => [17]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0
[8,8,1]
 => [16,1]
 => [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1
[6,4,4,3]
 => [8,3,3,3]
 => [4,4,4,1,1,1,1,1]
 => [4,4,1,1,1,1,1]
 => ? = 1
[4,4,3,3,3]
 => [8,6,3]
 => [3,3,3,2,2,2,1,1]
 => [3,3,2,2,2,1,1]
 => ? = 2
[]
 => ?
 => ?
 => ?
 => ? = 0
[6,5,4,3,2,1]
 => ?
 => ?
 => ?
 => ? = 0
[5,5,4,3,2,1]
 => ?
 => ?
 => ?
 => ? = 1
[6,4,4,3,2,1]
 => ?
 => ?
 => ?
 => ? = 1
[5,4,4,3,2,1]
 => ?
 => ?
 => ?
 => ? = 1
[4,4,4,3,2,1]
 => ?
 => ?
 => ?
 => ? = 1
[6,5,3,3,2,1]
 => ?
 => ?
 => ?
 => ? = 1
[5,5,3,3,2,1]
 => ?
 => ?
 => ?
 => ? = 2
[5,4,3,3,2,1]
 => ?
 => ?
 => ?
 => ? = 1
[6,3,3,3,2,1]
 => ?
 => ?
 => ?
 => ? = 1
[6,5,4,2,2,1]
 => ?
 => ?
 => ?
 => ? = 1
[5,5,4,2,2,1]
 => ?
 => ?
 => ?
 => ? = 2
[6,4,4,2,2,1]
 => ?
 => ?
 => ?
 => ? = 2
[6,5,2,2,2,1]
 => ?
 => ?
 => ?
 => ? = 1
[6,5,4,3,1,1]
 => ?
 => ?
 => ?
 => ? = 1
[5,5,4,3,1,1]
 => ?
 => ?
 => ?
 => ? = 2
[6,4,4,3,1,1]
 => ?
 => ?
 => ?
 => ? = 2
[6,5,3,3,1,1]
 => ?
 => ?
 => ?
 => ? = 2
[5,5,3,3,1,1]
 => ?
 => ?
 => ?
 => ? = 3
[6,5,4,1,1,1]
 => ?
 => ?
 => ?
 => ? = 1
[6,5,4,3,2]
 => ?
 => ?
 => ?
 => ? = 0
[5,5,4,3,2]
 => ?
 => ?
 => ?
 => ? = 1
[6,4,4,3,2]
 => ?
 => ?
 => ?
 => ? = 1
[6,5,3,3,2]
 => ?
 => ?
 => ?
 => ? = 1
[5,5,3,3,2]
 => ?
 => ?
 => ?
 => ? = 2
[6,5,4,2,2]
 => ?
 => ?
 => ?
 => ? = 1
[5,5,4,2,2]
 => ?
 => ?
 => ?
 => ? = 2
[6,4,4,2,2]
 => ?
 => ?
 => ?
 => ? = 2
[6,5,4,3,1]
 => ?
 => ?
 => ?
 => ? = 0
[6,5,4,3]
 => ?
 => ?
 => ?
 => ? = 0
[7,6,5,4,3,2,1]
 => ?
 => ?
 => ?
 => ? = 0
[6,6,5,4,3,2,1]
 => ?
 => ?
 => ?
 => ? = 1

Description

The alternating sum of the parts of an integer partition. For a partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ , this is

$\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$ .

Matching statistic: St000149
(load all 2 compositions to match this statistic)

Mapping

Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 57%●distinct values known / distinct values provided: 56%

Values

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

Description

The number of cells of the partition whose leg is zero and arm is odd. This statistic is equidistributed with [[St000143]], see [1].

Matching statistic: St000142
(load all 2 compositions to match this statistic)

Mapping

Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of even parts of a partition.

Matching statistic: St001487

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 17%

Values

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

Description

The number of inner corners of a skew partition.