Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001389
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => [1,1]
 => 1
[2]
 => 0 => [2] => [2]
 => 2
[1,1]
 => 11 => [1,1,1] => [1,1,1]
 => 1
[3]
 => 1 => [1,1] => [1,1]
 => 1
[2,1]
 => 01 => [2,1] => [2,1]
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 1
[4]
 => 0 => [2] => [2]
 => 2
[3,1]
 => 11 => [1,1,1] => [1,1,1]
 => 1
[2,2]
 => 00 => [3] => [3]
 => 3
[2,1,1]
 => 011 => [2,1,1] => [2,1,1]
 => 2
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
[5]
 => 1 => [1,1] => [1,1]
 => 1
[4,1]
 => 01 => [2,1] => [2,1]
 => 2
[3,2]
 => 10 => [1,2] => [2,1]
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 1
[2,2,1]
 => 001 => [3,1] => [3,1]
 => 3
[2,1,1,1]
 => 0111 => [2,1,1,1] => [2,1,1,1]
 => 2
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 1
[6]
 => 0 => [2] => [2]
 => 2
[5,1]
 => 11 => [1,1,1] => [1,1,1]
 => 1
[4,2]
 => 00 => [3] => [3]
 => 3
[4,1,1]
 => 011 => [2,1,1] => [2,1,1]
 => 2
[3,3]
 => 11 => [1,1,1] => [1,1,1]
 => 1
[3,2,1]
 => 101 => [1,2,1] => [2,1,1]
 => 2
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
[2,2,2]
 => 000 => [4] => [4]
 => 4
[2,2,1,1]
 => 0011 => [3,1,1] => [3,1,1]
 => 3
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
 => 1
[7]
 => 1 => [1,1] => [1,1]
 => 1
[6,1]
 => 01 => [2,1] => [2,1]
 => 2
[5,2]
 => 10 => [1,2] => [2,1]
 => 2
[5,1,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 1
[4,3]
 => 01 => [2,1] => [2,1]
 => 2
[4,2,1]
 => 001 => [3,1] => [3,1]
 => 3
[4,1,1,1]
 => 0111 => [2,1,1,1] => [2,1,1,1]
 => 2
[3,3,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 1
[3,2,2]
 => 100 => [1,3] => [3,1]
 => 3
[3,2,1,1]
 => 1011 => [1,2,1,1] => [2,1,1,1]
 => 2
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 1
[2,2,2,1]
 => 0001 => [4,1] => [4,1]
 => 4
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => [3,1,1,1]
 => 3
[2,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
 => 1
[8]
 => 0 => [2] => [2]
 => 2
[7,1]
 => 11 => [1,1,1] => [1,1,1]
 => 1
[6,2]
 => 00 => [3] => [3]
 => 3
[6,1,1]
 => 011 => [2,1,1] => [2,1,1]
 => 2
[5,3]
 => 11 => [1,1,1] => [1,1,1]
 => 1
[5,2,1]
 => 101 => [1,2,1] => [2,1,1]
 => 2
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000389
(load all 2 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 70%
Values
[1]
 => [[1]]
 => [[1]]
 => => ? = 1 - 1
[2]
 => [[1,2]]
 => [[1],[2]]
 => 1 => 1 = 2 - 1
[1,1]
 => [[1],[2]]
 => [[1,2]]
 => 0 => 0 = 1 - 1
[3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 11 => 0 = 1 - 1
[2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 01 => 1 = 2 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 00 => 0 = 1 - 1
[4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 111 => 1 = 2 - 1
[3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 011 => 0 = 1 - 1
[2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 101 => 2 = 3 - 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 001 => 1 = 2 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 000 => 0 = 1 - 1
[5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 1111 => 0 = 1 - 1
[4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 0111 => 1 = 2 - 1
[3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 1011 => 1 = 2 - 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 0011 => 0 = 1 - 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 0101 => 2 = 3 - 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 0001 => 1 = 2 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 0000 => 0 = 1 - 1
[6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 11111 => 1 = 2 - 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => 01111 => 0 = 1 - 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 10111 => 2 = 3 - 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [[1,2,3],[4],[5],[6]]
 => 00111 => 1 = 2 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 11011 => 0 = 1 - 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [[1,2,4],[3,5],[6]]
 => 01011 => 1 = 2 - 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [[1,2,3,4],[5],[6]]
 => 00011 => 0 = 1 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 10101 => 3 = 4 - 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 00101 => 2 = 3 - 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 00001 => 1 = 2 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 00000 => 0 = 1 - 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 111111 => 0 = 1 - 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => 011111 => 1 = 2 - 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 101111 => 1 = 2 - 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [[1,2,3],[4],[5],[6],[7]]
 => 001111 => 0 = 1 - 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7]]
 => 110111 => 1 = 2 - 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [[1,2,4],[3,5],[6],[7]]
 => 010111 => 2 = 3 - 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [[1,2,3,4],[5],[6],[7]]
 => 000111 => 1 = 2 - 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [[1,2,5],[3,6],[4,7]]
 => 011011 => 0 = 1 - 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => 101011 => 2 = 3 - 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6],[7]]
 => 001011 => 1 = 2 - 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6],[7]]
 => 000011 => 0 = 1 - 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [[1,2,4,6],[3,5,7]]
 => 010101 => 3 = 4 - 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [[1,2,3,4,6],[5,7]]
 => 000101 => 2 = 3 - 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6],[7]]
 => 000001 => 1 = 2 - 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[1,2,3,4,5,6,7]]
 => 000000 => 0 = 1 - 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 1111111 => 1 = 2 - 1
[7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => 0111111 => 0 = 1 - 1
[6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 1011111 => 2 = 3 - 1
[6,1,1]
 => [[1,4,5,6,7,8],[2],[3]]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => 0011111 => 1 = 2 - 1
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7],[8]]
 => 1101111 => 0 = 1 - 1
[5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [[1,2,4],[3,5],[6],[7],[8]]
 => 0101111 => 1 = 2 - 1
[5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => 0001111 => 0 = 1 - 1
[11]
 => [[1,2,3,4,5,6,7,8,9,10,11]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? => ? = 1 - 1
[10,1]
 => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? => ? = 2 - 1
[9,2]
 => [[1,2,5,6,7,8,9,10,11],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11]]
 => ? => ? = 2 - 1
[9,1,1]
 => [[1,4,5,6,7,8,9,10,11],[2],[3]]
 => ?
 => ? => ? = 1 - 1
[8,3]
 => [[1,2,3,7,8,9,10,11],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7],[8],[9],[10],[11]]
 => ? => ? = 2 - 1
[8,2,1]
 => [[1,3,6,7,8,9,10,11],[2,5],[4]]
 => ?
 => ? => ? = 3 - 1
[8,1,1,1]
 => [[1,5,6,7,8,9,10,11],[2],[3],[4]]
 => ?
 => ? => ? = 2 - 1
[7,4]
 => [[1,2,3,4,9,10,11],[5,6,7,8]]
 => ?
 => ? => ? = 2 - 1
[7,3,1]
 => [[1,3,4,8,9,10,11],[2,6,7],[5]]
 => ?
 => ? => ? = 1 - 1
[7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8],[9],[10],[11]]
 => ? => ? = 3 - 1
[7,2,1,1]
 => [[1,4,7,8,9,10,11],[2,6],[3],[5]]
 => ?
 => ? => ? = 2 - 1
[7,1,1,1,1]
 => [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
 => ?
 => ? => ? = 1 - 1
[6,5]
 => [[1,2,3,4,5,11],[6,7,8,9,10]]
 => [[1,6],[2,7],[3,8],[4,9],[5,10],[11]]
 => ? => ? = 2 - 1
[6,4,1]
 => [[1,3,4,5,10,11],[2,7,8,9],[6]]
 => ?
 => ? => ? = 3 - 1
[6,3,2]
 => [[1,2,5,9,10,11],[3,4,8],[6,7]]
 => ?
 => ? => ? = 3 - 1
[6,3,1,1]
 => [[1,4,5,9,10,11],[2,7,8],[3],[6]]
 => ?
 => ? => ? = 2 - 1
[6,2,2,1]
 => [[1,3,8,9,10,11],[2,5],[4,7],[6]]
 => ?
 => ? => ? = 4 - 1
[6,2,1,1,1]
 => [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
 => [[1,2,3,4,6],[5,7],[8],[9],[10],[11]]
 => ? => ? = 3 - 1
[6,1,1,1,1,1]
 => [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
 => ?
 => ? => ? = 2 - 1
[5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => [[1,2,7],[3,8],[4,9],[5,10],[6,11]]
 => ? => ? = 1 - 1
[5,4,2]
 => [[1,2,5,6,11],[3,4,9,10],[7,8]]
 => [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
 => 1011101111 => ? = 3 - 1
[5,4,1,1]
 => [[1,4,5,6,11],[2,8,9,10],[3],[7]]
 => [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
 => 0011101111 => ? = 2 - 1
[5,3,3]
 => [[1,2,3,10,11],[4,5,6],[7,8,9]]
 => [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
 => 1101101111 => ? = 1 - 1
[5,3,2,1]
 => [[1,3,6,10,11],[2,5,9],[4,8],[7]]
 => [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
 => 0101101111 => ? = 2 - 1
[5,2,2,2]
 => [[1,2,9,10,11],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
 => 1010101111 => ? = 4 - 1
[5,2,1,1,1,1]
 => [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
 => [[1,2,3,4,5,7],[6,8],[9],[10],[11]]
 => ? => ? = 2 - 1
[5,1,1,1,1,1,1]
 => [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
 => ?
 => ? => ? = 1 - 1
[4,4,3]
 => [[1,2,3,7],[4,5,6,11],[8,9,10]]
 => [[1,4,8],[2,5,9],[3,6,10],[7,11]]
 => 1101110111 => ? = 3 - 1
[4,4,2,1]
 => [[1,3,6,7],[2,5,10,11],[4,9],[8]]
 => [[1,2,4,8],[3,5,9],[6,10],[7,11]]
 => 0101110111 => ? = 4 - 1
[4,3,3,1]
 => [[1,3,4,11],[2,6,7],[5,9,10],[8]]
 => [[1,2,5,8],[3,6,9],[4,7,10],[11]]
 => 0110110111 => ? = 2 - 1
[4,3,2,2]
 => [[1,2,7,11],[3,4,10],[5,6],[8,9]]
 => [[1,3,5,8],[2,4,6,9],[7,10],[11]]
 => 1010110111 => ? = 5 - 1
[4,3,2,1,1]
 => [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
 => [[1,2,3,5,8],[4,6,9],[7,10],[11]]
 => 0010110111 => ? = 3 - 1
[4,3,1,1,1,1]
 => [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
 => ?
 => ? => ? = 2 - 1
[4,2,2,2,1]
 => [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
 => [[1,2,4,6,8],[3,5,7,9],[10],[11]]
 => 0101010111 => ? = 5 - 1
[4,2,2,1,1,1]
 => [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
 => ?
 => ? => ? = 4 - 1
[4,2,1,1,1,1,1]
 => [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
 => ?
 => ? => ? = 3 - 1
[4,1,1,1,1,1,1,1]
 => [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
 => ?
 => ? => ? = 2 - 1
[3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11]]
 => 1011011011 => ? = 2 - 1
[3,3,3,1,1]
 => [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
 => [[1,2,3,6,9],[4,7,10],[5,8,11]]
 => 0011011011 => ? = 1 - 1
[3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
 => [[1,2,4,6,9],[3,5,7,10],[8,11]]
 => 0101011011 => ? = 3 - 1
[3,3,2,1,1,1]
 => [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
 => ?
 => ? => ? = 2 - 1
[3,3,1,1,1,1,1]
 => [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
 => [[1,2,3,4,5,6,9],[7,10],[8,11]]
 => ? => ? = 1 - 1
[3,2,2,2,2]
 => [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
 => [[1,3,5,7,9],[2,4,6,8,10],[11]]
 => ? => ? = 5 - 1
[3,2,2,2,1,1]
 => [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
 => ?
 => ? => ? = 4 - 1
[3,2,2,1,1,1,1]
 => [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
 => ?
 => ? => ? = 3 - 1
[3,2,1,1,1,1,1,1]
 => [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
 => ?
 => ? => ? = 2 - 1
[3,1,1,1,1,1,1,1,1]
 => [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ?
 => ? => ? = 1 - 1
[2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
 => [[1,2,4,6,8,10],[3,5,7,9,11]]
 => ? => ? = 6 - 1
[2,2,2,2,1,1,1]
 => [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
 => ?
 => ? => ? = 5 - 1
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000237
(load all 2 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 70%
Values
[1]
 => [[1]]
 => [1] => [1] => 0 = 1 - 1
[2]
 => [[1,2]]
 => [1,2] => [2,1] => 1 = 2 - 1
[1,1]
 => [[1],[2]]
 => [2,1] => [1,2] => 0 = 1 - 1
[3]
 => [[1,2,3]]
 => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => [2,1,3] => 1 = 2 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => 1 = 2 - 1
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => [3,2,1,4] => 0 = 1 - 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [2,1,3,4] => 1 = 2 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [4,3,2,1,5] => 1 = 2 - 1
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [3,2,1,5,4] => 1 = 2 - 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [3,2,1,4,5] => 0 = 1 - 1
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [2,1,4,3,5] => 2 = 3 - 1
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 1 = 2 - 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [5,4,3,2,1,6] => 0 = 1 - 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 2 = 3 - 1
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [4,3,2,1,5,6] => 1 = 2 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 0 = 1 - 1
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => [3,2,1,5,4,6] => 1 = 2 - 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [3,2,1,4,5,6] => 0 = 1 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 3 = 4 - 1
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => [2,1,4,3,5,6] => 2 = 3 - 1
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [2,1,3,4,5,6] => 1 = 2 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 0 = 1 - 1
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => 1 = 2 - 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => 1 = 2 - 1
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => 0 = 1 - 1
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => ? = 2 - 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => ? = 3 - 1
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => 1 = 2 - 1
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => [3,2,1,6,5,4,7] => ? = 1 - 1
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => [3,2,1,5,4,7,6] => ? = 3 - 1
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => [3,2,1,5,4,6,7] => ? = 2 - 1
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => 0 = 1 - 1
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [2,1,4,3,6,5,7] => ? = 4 - 1
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => [2,1,4,3,5,6,7] => ? = 3 - 1
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => [2,1,3,4,5,6,7] => 1 = 2 - 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 1 = 2 - 1
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => 0 = 1 - 1
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => 2 = 3 - 1
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => 1 = 2 - 1
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => 0 = 1 - 1
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => ? = 2 - 1
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => 0 = 1 - 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => 2 = 3 - 1
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => ? = 2 - 1
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => 3 = 4 - 1
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => ? = 3 - 1
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => 1 = 2 - 1
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ? = 2 - 1
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => ? = 1 - 1
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => ? = 3 - 1
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ? = 2 - 1
[3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => 0 = 1 - 1
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => 4 = 5 - 1
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => ? = 4 - 1
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => ? = 3 - 1
[2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => 1 = 2 - 1
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8] => ? = 2 - 1
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7] => ? = 2 - 1
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9] => ? = 3 - 1
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6] => ? = 2 - 1
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,9] => ? = 1 - 1
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8] => ? = 3 - 1
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8,9] => ? = 2 - 1
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9] => ? = 3 - 1
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8] => ? = 3 - 1
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8,9] => ? = 2 - 1
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,9] => ? = 4 - 1
[4,2,1,1,1]
 => [[1,2,3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8,9] => ? = 3 - 1
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => [3,2,1,6,5,4,9,8,7] => ? = 1 - 1
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7,9] => ? = 2 - 1
[3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8,9] => ? = 1 - 1
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,9,8] => ? = 4 - 1
[3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [9,8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8,9] => ? = 3 - 1
[3,2,1,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8],[9]]
 => [9,8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8,9] => ? = 2 - 1
[2,2,2,2,1]
 => [[1,2],[3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7,9] => ? = 5 - 1
[2,2,2,1,1,1]
 => [[1,2],[3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8,9] => ? = 4 - 1
[2,2,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8,9] => ? = 3 - 1
[7,3]
 => [[1,2,3,4,5,6,7],[8,9,10]]
 => [8,9,10,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,10,9,8] => ? = 1 - 1
[7,2,1]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => [10,8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8,10] => ? = 2 - 1
[6,3,1]
 => [[1,2,3,4,5,6],[7,8,9],[10]]
 => [10,7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7,10] => ? = 2 - 1
[6,2,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10]]
 => [10,9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9,10] => ? = 3 - 1
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [6,7,8,9,10,1,2,3,4,5] => [5,4,3,2,1,10,9,8,7,6] => ? = 1 - 1
[5,4,1]
 => [[1,2,3,4,5],[6,7,8,9],[10]]
 => [10,6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6,10] => ? = 2 - 1
[5,3,2]
 => [[1,2,3,4,5],[6,7,8],[9,10]]
 => [9,10,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,10,9] => ? = 2 - 1
[5,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10]]
 => [10,9,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,9,10] => ? = 1 - 1
[5,2,2,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10]]
 => [10,8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8,10] => ? = 3 - 1
[5,2,1,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9],[10]]
 => [10,9,8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8,9,10] => ? = 2 - 1
[4,4,1,1]
 => [[1,2,3,4],[5,6,7,8],[9],[10]]
 => [10,9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9,10] => ? = 3 - 1
[4,3,3]
 => [[1,2,3,4],[5,6,7],[8,9,10]]
 => [8,9,10,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,10,9,8] => ? = 2 - 1
[4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8,10] => ? = 3 - 1
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St001465
(load all 2 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 70%
Values
[1]
 => [[1]]
 => [1] => [1] => 0 = 1 - 1
[2]
 => [[1,2]]
 => [1,2] => [2,1] => 1 = 2 - 1
[1,1]
 => [[1],[2]]
 => [2,1] => [1,2] => 0 = 1 - 1
[3]
 => [[1,2,3]]
 => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => [2,1,3] => 1 = 2 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => 1 = 2 - 1
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => [3,2,1,4] => 0 = 1 - 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [2,1,3,4] => 1 = 2 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [4,3,2,1,5] => 1 = 2 - 1
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [3,2,1,5,4] => 1 = 2 - 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [3,2,1,4,5] => 0 = 1 - 1
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [2,1,4,3,5] => 2 = 3 - 1
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 1 = 2 - 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [5,4,3,2,1,6] => 0 = 1 - 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 2 = 3 - 1
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [4,3,2,1,5,6] => 1 = 2 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 0 = 1 - 1
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => [3,2,1,5,4,6] => 1 = 2 - 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [3,2,1,4,5,6] => 0 = 1 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 3 = 4 - 1
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => [2,1,4,3,5,6] => 2 = 3 - 1
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [2,1,3,4,5,6] => 1 = 2 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 0 = 1 - 1
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => 1 = 2 - 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => 1 = 2 - 1
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => 0 = 1 - 1
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => ? = 2 - 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => ? = 3 - 1
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => 1 = 2 - 1
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => [3,2,1,6,5,4,7] => ? = 1 - 1
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => [3,2,1,5,4,7,6] => ? = 3 - 1
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => [3,2,1,5,4,6,7] => ? = 2 - 1
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => 0 = 1 - 1
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [2,1,4,3,6,5,7] => ? = 4 - 1
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => [2,1,4,3,5,6,7] => ? = 3 - 1
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => [2,1,3,4,5,6,7] => 1 = 2 - 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 1 = 2 - 1
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => 0 = 1 - 1
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => 2 = 3 - 1
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => 1 = 2 - 1
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => 0 = 1 - 1
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => ? = 2 - 1
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => 0 = 1 - 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => 2 = 3 - 1
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => ? = 2 - 1
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => 3 = 4 - 1
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => ? = 3 - 1
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => 1 = 2 - 1
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ? = 2 - 1
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => ? = 1 - 1
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => ? = 3 - 1
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ? = 2 - 1
[3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => 0 = 1 - 1
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => 4 = 5 - 1
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => ? = 4 - 1
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => ? = 3 - 1
[2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => 1 = 2 - 1
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8] => ? = 2 - 1
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7] => ? = 2 - 1
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9] => ? = 3 - 1
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6] => ? = 2 - 1
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,9] => ? = 1 - 1
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8] => ? = 3 - 1
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8,9] => ? = 2 - 1
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9] => ? = 3 - 1
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8] => ? = 3 - 1
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8,9] => ? = 2 - 1
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,9] => ? = 4 - 1
[4,2,1,1,1]
 => [[1,2,3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8,9] => ? = 3 - 1
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => [3,2,1,6,5,4,9,8,7] => ? = 1 - 1
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7,9] => ? = 2 - 1
[3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8,9] => ? = 1 - 1
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,9,8] => ? = 4 - 1
[3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [9,8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8,9] => ? = 3 - 1
[3,2,1,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8],[9]]
 => [9,8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8,9] => ? = 2 - 1
[2,2,2,2,1]
 => [[1,2],[3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7,9] => ? = 5 - 1
[2,2,2,1,1,1]
 => [[1,2],[3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8,9] => ? = 4 - 1
[2,2,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9]]
 => [9,8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8,9] => ? = 3 - 1
[7,3]
 => [[1,2,3,4,5,6,7],[8,9,10]]
 => [8,9,10,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,10,9,8] => ? = 1 - 1
[7,2,1]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => [10,8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8,10] => ? = 2 - 1
[6,3,1]
 => [[1,2,3,4,5,6],[7,8,9],[10]]
 => [10,7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7,10] => ? = 2 - 1
[6,2,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10]]
 => [10,9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9,10] => ? = 3 - 1
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [6,7,8,9,10,1,2,3,4,5] => [5,4,3,2,1,10,9,8,7,6] => ? = 1 - 1
[5,4,1]
 => [[1,2,3,4,5],[6,7,8,9],[10]]
 => [10,6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6,10] => ? = 2 - 1
[5,3,2]
 => [[1,2,3,4,5],[6,7,8],[9,10]]
 => [9,10,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,10,9] => ? = 2 - 1
[5,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10]]
 => [10,9,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,9,10] => ? = 1 - 1
[5,2,2,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10]]
 => [10,8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8,10] => ? = 3 - 1
[5,2,1,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9],[10]]
 => [10,9,8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8,9,10] => ? = 2 - 1
[4,4,1,1]
 => [[1,2,3,4],[5,6,7,8],[9],[10]]
 => [10,9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9,10] => ? = 3 - 1
[4,3,3]
 => [[1,2,3,4],[5,6,7],[8,9,10]]
 => [8,9,10,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,10,9,8] => ? = 2 - 1
[4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8,10] => ? = 3 - 1
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St001232
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 70%
Values
[1]
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
[2]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,1]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 1 - 1
[2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1 - 1
[2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[3,2]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2 - 1
[3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1 - 1
[2,2,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[2,1,1,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[4,2]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 3 - 1
[4,1,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 1 - 1
[3,2,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2 - 1
[3,1,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1 - 1
[2,2,2]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,1,1,1,1]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,1,1]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[6,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[5,2]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[5,1,1]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[4,3]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 2 - 1
[4,2,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => ? = 3 - 1
[4,1,1,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[3,3,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 1 - 1
[3,2,2]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 3 - 1
[3,2,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 2 - 1
[3,1,1,1,1]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1 - 1
[2,2,2,1]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[2,2,1,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,1,1,1,1,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,1,1,1]
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
[8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[7,1]
 => [2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[6,2]
 => [2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 3 - 1
[6,1,1]
 => [3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[5,3]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 1 - 1
[5,2,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[5,1,1,1]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[4,4]
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 3 - 1
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => ? = 2 - 1
[4,2,2]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 4 - 1
[4,2,1,1]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 3 - 1
[4,1,1,1,1]
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[3,3,2]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 2 - 1
[3,3,1,1]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 1 - 1
[3,2,2,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ? = 3 - 1
[3,2,1,1,1]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => ? = 2 - 1
[3,1,1,1,1,1]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1 - 1
[2,2,2,2]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 5 - 1
[2,2,2,1,1]
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3 = 4 - 1
[2,2,1,1,1,1]
 => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,1,1,1,1,1,1]
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,1,1,1,1,1,1,1]
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1 - 1
[9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[8,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[7,2]
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[7,1,1]
 => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[6,3]
 => [2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[6,2,1]
 => [3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 3 - 1
[2,2,2,2,1]
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4 = 5 - 1
[2,2,2,1,1,1]
 => [6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3 = 4 - 1
[2,2,2,2,2]
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[2,2,2,2,1,1]
 => [6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4 = 5 - 1
[2,2,2,2,2,1]
 => [6,5]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[2,2,2,2,2,2]
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6 = 7 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.