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Matching statistic: St001124
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 1
[2]
=> 100 => [1,3] => [3,1]
=> 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 1
[3]
=> 1000 => [1,4] => [4,1]
=> 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 1
[4]
=> 10000 => [1,5] => [5,1]
=> 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 2
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 1
[5]
=> 100000 => [1,6] => [6,1]
=> 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 2
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 2
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 2
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 2
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 2
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 2
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 2
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 2
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 2
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 2
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 2
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 2
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 2
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 2
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 2
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 2
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 2
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 2
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 2
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 2
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000159
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 2 = 1 + 1
[2]
=> 100 => [1,3] => [3,1]
=> 2 = 1 + 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 2 = 1 + 1
[3]
=> 1000 => [1,4] => [4,1]
=> 2 = 1 + 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 2 = 1 + 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 2 = 1 + 1
[4]
=> 10000 => [1,5] => [5,1]
=> 2 = 1 + 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 3 = 2 + 1
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 2 = 1 + 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[5]
=> 100000 => [1,6] => [6,1]
=> 2 = 1 + 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 3 = 2 + 1
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 3 = 2 + 1
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 3 = 2 + 1
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 2 = 1 + 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 2 = 1 + 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 3 = 2 + 1
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 2 = 1 + 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 3 = 2 + 1
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 2 = 1 + 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 2 = 1 + 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 3 = 2 + 1
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 2 = 1 + 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 2 = 1 + 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 2 = 1 + 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 2 = 1 + 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 3 = 2 + 1
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 3 = 2 + 1
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 3 = 2 + 1
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 3 = 2 + 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 3 = 2 + 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 3 = 2 + 1
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 3 = 2 + 1
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 3 = 2 + 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 3 = 2 + 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 2 = 1 + 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 2 = 1 + 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 2 = 1 + 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 3 = 2 + 1
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 3 = 2 + 1
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 3 = 2 + 1
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 3 = 2 + 1
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 3 = 2 + 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 3 = 1 + 2
[2]
=> 100 => [1,3] => [3,1]
=> 3 = 1 + 2
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 3 = 1 + 2
[3]
=> 1000 => [1,4] => [4,1]
=> 3 = 1 + 2
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 3 = 1 + 2
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 3 = 1 + 2
[4]
=> 10000 => [1,5] => [5,1]
=> 3 = 1 + 2
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 4 = 2 + 2
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 3 = 1 + 2
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 3 = 1 + 2
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 3 = 1 + 2
[5]
=> 100000 => [1,6] => [6,1]
=> 3 = 1 + 2
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 4 = 2 + 2
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 4 = 2 + 2
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 4 = 2 + 2
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 3 = 1 + 2
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 3 = 1 + 2
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 3 = 1 + 2
[6]
=> 1000000 => [1,7] => [7,1]
=> 3 = 1 + 2
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 4 = 2 + 2
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 3 = 1 + 2
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 4 = 2 + 2
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 3 = 1 + 2
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 3 = 1 + 2
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 4 = 2 + 2
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 3 = 1 + 2
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 3 = 1 + 2
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 3 = 1 + 2
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 3 = 1 + 2
[7]
=> 10000000 => [1,8] => [8,1]
=> 3 = 1 + 2
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 4 = 2 + 2
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 4 = 2 + 2
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 4 = 2 + 2
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 4 = 2 + 2
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 4 = 2 + 2
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 4 = 2 + 2
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 4 = 2 + 2
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 4 = 2 + 2
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 3 = 1 + 2
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 4 = 2 + 2
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 3 = 1 + 2
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 3 = 1 + 2
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 3 = 1 + 2
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 3 = 1 + 2
[8]
=> 100000000 => [1,9] => [9,1]
=> 3 = 1 + 2
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 4 = 2 + 2
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 4 = 2 + 2
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 4 = 2 + 2
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 4 = 2 + 2
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 4 = 2 + 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000897
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000897: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
St000897: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1
[2]
=> [1,1]
=> 1
[1,1]
=> [2]
=> 1
[3]
=> [1,1,1]
=> 1
[2,1]
=> [2,1]
=> 1
[1,1,1]
=> [3]
=> 1
[4]
=> [1,1,1,1]
=> 1
[3,1]
=> [2,1,1]
=> 2
[2,2]
=> [2,2]
=> 1
[2,1,1]
=> [3,1]
=> 1
[1,1,1,1]
=> [4]
=> 1
[5]
=> [1,1,1,1,1]
=> 1
[4,1]
=> [2,1,1,1]
=> 2
[3,2]
=> [2,2,1]
=> 2
[3,1,1]
=> [3,1,1]
=> 2
[2,2,1]
=> [3,2]
=> 1
[2,1,1,1]
=> [4,1]
=> 1
[1,1,1,1,1]
=> [5]
=> 1
[6]
=> [1,1,1,1,1,1]
=> 1
[5,1]
=> [2,1,1,1,1]
=> 2
[4,2]
=> [2,2,1,1]
=> 1
[4,1,1]
=> [3,1,1,1]
=> 2
[3,3]
=> [2,2,2]
=> 1
[3,2,1]
=> [3,2,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> 2
[2,2,2]
=> [3,3]
=> 1
[2,2,1,1]
=> [4,2]
=> 1
[2,1,1,1,1]
=> [5,1]
=> 1
[1,1,1,1,1,1]
=> [6]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> 1
[6,1]
=> [2,1,1,1,1,1]
=> 2
[5,2]
=> [2,2,1,1,1]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> 2
[4,3]
=> [2,2,2,1]
=> 2
[4,2,1]
=> [3,2,1,1]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> 2
[3,3,1]
=> [3,2,2]
=> 2
[3,2,2]
=> [3,3,1]
=> 2
[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]
=> 1
[2,1,1,1,1,1]
=> [6,1]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> 1
[8]
=> [1,1,1,1,1,1,1,1]
=> 1
[7,1]
=> [2,1,1,1,1,1,1]
=> 2
[6,2]
=> [2,2,1,1,1,1]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> 2
[5,3]
=> [2,2,2,1,1]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> 2
[8,4,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 3
[7,6]
=> [2,2,2,2,2,2,1]
=> ? = 2
[6,6,1]
=> [3,2,2,2,2,2]
=> ? = 2
[6,5,2]
=> [3,3,2,2,2,1]
=> ? = 3
[6,4,3]
=> [3,3,3,2,1,1]
=> ? = 3
[5,5,3]
=> [3,3,3,2,2]
=> ? = 2
[5,5,2,1]
=> [4,3,2,2,2]
=> ? = 2
[5,4,4]
=> [3,3,3,3,1]
=> ? = 2
[5,4,3,1]
=> [4,3,3,2,1]
=> ? = 2
[5,4,2,2]
=> [4,4,2,2,1]
=> ? = 2
[5,3,3,2]
=> [4,4,3,1,1]
=> ? = 2
[4,4,4,1]
=> [4,3,3,3]
=> ? = 2
[4,4,3,2]
=> [4,4,3,2]
=> ? = 2
[4,4,3,1,1]
=> [5,3,3,2]
=> ? = 2
[4,4,2,2,1]
=> [5,4,2,2]
=> ? = 2
[4,3,3,3]
=> [4,4,4,1]
=> ? = 2
[4,3,3,2,1]
=> [5,4,3,1]
=> ? = 1
[4,3,2,2,2]
=> [5,5,2,1]
=> ? = 2
[3,3,3,3,1]
=> [5,4,4]
=> ? = 2
[3,3,3,2,2]
=> [5,5,3]
=> ? = 2
[3,3,3,2,1,1]
=> [6,4,3]
=> ? = 1
[3,3,2,2,2,1]
=> [6,5,2]
=> ? = 1
[3,2,2,2,2,2]
=> [6,6,1]
=> ? = 2
[2,2,2,2,2,2,1]
=> [7,6]
=> ? = 1
[8,5,1]
=> [3,2,2,2,2,1,1,1]
=> ? = 3
[8,4,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 2
[7,7]
=> [2,2,2,2,2,2,2]
=> ? = 1
[7,4,2,1]
=> [4,3,2,2,1,1,1]
=> ? = 3
[6,6,2]
=> [3,3,2,2,2,2]
=> ? = 2
[6,5,3]
=> [3,3,3,2,2,1]
=> ? = 3
[6,4,4]
=> [3,3,3,3,1,1]
=> ? = 2
[5,5,4]
=> [3,3,3,3,2]
=> ? = 2
[5,5,3,1]
=> [4,3,3,2,2]
=> ? = 2
[5,5,2,2]
=> [4,4,2,2,2]
=> ? = 2
[5,4,4,1]
=> [4,3,3,3,1]
=> ? = 2
[5,4,3,2]
=> [4,4,3,2,1]
=> ? = 2
[5,3,3,3]
=> [4,4,4,1,1]
=> ? = 2
[4,4,4,2]
=> [4,4,3,3]
=> ? = 1
[4,4,4,1,1]
=> [5,3,3,3]
=> ? = 2
[4,4,3,3]
=> [4,4,4,2]
=> ? = 2
[4,4,3,2,1]
=> [5,4,3,2]
=> ? = 1
[4,4,2,2,2]
=> [5,5,2,2]
=> ? = 1
[4,3,3,3,1]
=> [5,4,4,1]
=> ? = 2
[4,3,3,2,2]
=> [5,5,3,1]
=> ? = 2
[3,3,3,3,2]
=> [5,5,4]
=> ? = 2
[3,3,3,3,1,1]
=> [6,4,4]
=> ? = 2
[3,3,3,2,2,1]
=> [6,5,3]
=> ? = 1
[3,3,2,2,2,2]
=> [6,6,2]
=> ? = 2
[2,2,2,2,2,2,2]
=> [7,7]
=> ? = 1
[8,5,2]
=> [3,3,2,2,2,1,1,1]
=> ? = 2
Description
The number of different multiplicities of parts of an integer partition.
Matching statistic: St000903
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%
Mp00178: Binary words —to composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%
Values
[1]
=> 10 => [1,2] => 2 = 1 + 1
[2]
=> 100 => [1,3] => 2 = 1 + 1
[1,1]
=> 110 => [1,1,2] => 2 = 1 + 1
[3]
=> 1000 => [1,4] => 2 = 1 + 1
[2,1]
=> 1010 => [1,2,2] => 2 = 1 + 1
[1,1,1]
=> 1110 => [1,1,1,2] => 2 = 1 + 1
[4]
=> 10000 => [1,5] => 2 = 1 + 1
[3,1]
=> 10010 => [1,3,2] => 3 = 2 + 1
[2,2]
=> 1100 => [1,1,3] => 2 = 1 + 1
[2,1,1]
=> 10110 => [1,2,1,2] => 2 = 1 + 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => 2 = 1 + 1
[5]
=> 100000 => [1,6] => 2 = 1 + 1
[4,1]
=> 100010 => [1,4,2] => 3 = 2 + 1
[3,2]
=> 10100 => [1,2,3] => 3 = 2 + 1
[3,1,1]
=> 100110 => [1,3,1,2] => 3 = 2 + 1
[2,2,1]
=> 11010 => [1,1,2,2] => 2 = 1 + 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => 2 = 1 + 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => 2 = 1 + 1
[6]
=> 1000000 => [1,7] => 2 = 1 + 1
[5,1]
=> 1000010 => [1,5,2] => 3 = 2 + 1
[4,2]
=> 100100 => [1,3,3] => 2 = 1 + 1
[4,1,1]
=> 1000110 => [1,4,1,2] => 3 = 2 + 1
[3,3]
=> 11000 => [1,1,4] => 2 = 1 + 1
[3,2,1]
=> 101010 => [1,2,2,2] => 2 = 1 + 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => 3 = 2 + 1
[2,2,2]
=> 11100 => [1,1,1,3] => 2 = 1 + 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => 2 = 1 + 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => 2 = 1 + 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => 2 = 1 + 1
[7]
=> 10000000 => [1,8] => 2 = 1 + 1
[6,1]
=> 10000010 => [1,6,2] => 3 = 2 + 1
[5,2]
=> 1000100 => [1,4,3] => 3 = 2 + 1
[5,1,1]
=> 10000110 => [1,5,1,2] => 3 = 2 + 1
[4,3]
=> 101000 => [1,2,4] => 3 = 2 + 1
[4,2,1]
=> 1001010 => [1,3,2,2] => 3 = 2 + 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => 3 = 2 + 1
[3,3,1]
=> 110010 => [1,1,3,2] => 3 = 2 + 1
[3,2,2]
=> 101100 => [1,2,1,3] => 3 = 2 + 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => 3 = 2 + 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => 2 = 1 + 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => 2 = 1 + 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => 2 = 1 + 1
[8]
=> 100000000 => [1,9] => ? = 1 + 1
[7,1]
=> 100000010 => [1,7,2] => ? = 2 + 1
[6,2]
=> 10000100 => [1,5,3] => 3 = 2 + 1
[6,1,1]
=> 100000110 => [1,6,1,2] => ? = 2 + 1
[5,3]
=> 1001000 => [1,3,4] => 3 = 2 + 1
[5,2,1]
=> 10001010 => [1,4,2,2] => 3 = 2 + 1
[5,1,1,1]
=> 100001110 => [1,5,1,1,2] => ? = 2 + 1
[4,4]
=> 110000 => [1,1,5] => 2 = 1 + 1
[4,3,1]
=> 1010010 => [1,2,3,2] => 3 = 2 + 1
[4,2,2]
=> 1001100 => [1,3,1,3] => 2 = 1 + 1
[4,1,1,1,1]
=> 100011110 => [1,4,1,1,1,2] => ? = 2 + 1
[3,1,1,1,1,1]
=> 100111110 => [1,3,1,1,1,1,2] => ? = 2 + 1
[2,1,1,1,1,1,1]
=> 101111110 => [1,2,1,1,1,1,1,2] => ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> 111111110 => [1,1,1,1,1,1,1,1,2] => ? = 1 + 1
[7,2]
=> 100000100 => [1,6,3] => ? = 2 + 1
[6,2,1]
=> 100001010 => [1,5,2,2] => ? = 2 + 1
[5,2,1,1]
=> 100010110 => [1,4,2,1,2] => ? = 2 + 1
[4,2,1,1,1]
=> 100101110 => [1,3,2,1,1,2] => ? = 2 + 1
[3,2,1,1,1,1]
=> 101011110 => [1,2,2,1,1,1,2] => ? = 1 + 1
[2,2,1,1,1,1,1]
=> 110111110 => [1,1,2,1,1,1,1,2] => ? = 1 + 1
[7,3]
=> 100001000 => [1,5,4] => ? = 2 + 1
[6,3,1]
=> 100010010 => [1,4,3,2] => ? = 3 + 1
[6,2,2]
=> 100001100 => [1,5,1,3] => ? = 2 + 1
[6,2,1,1]
=> 1000010110 => [1,5,2,1,2] => ? = 2 + 1
[5,3,1,1]
=> 100100110 => [1,3,3,1,2] => ? = 2 + 1
[5,2,2,1]
=> 100011010 => [1,4,1,2,2] => ? = 2 + 1
[4,3,1,1,1]
=> 101001110 => [1,2,3,1,1,2] => ? = 2 + 1
[4,2,2,1,1]
=> 100110110 => [1,3,1,2,1,2] => ? = 2 + 1
[3,3,1,1,1,1]
=> 110011110 => [1,1,3,1,1,1,2] => ? = 2 + 1
[3,2,2,1,1,1]
=> 101101110 => [1,2,1,2,1,1,2] => ? = 1 + 1
[2,2,2,1,1,1,1]
=> 111011110 => [1,1,1,2,1,1,1,2] => ? = 1 + 1
[7,4]
=> 100010000 => [1,4,5] => ? = 2 + 1
[6,4,1]
=> 100100010 => [1,3,4,2] => ? = 3 + 1
[6,3,2]
=> 100010100 => [1,4,2,3] => ? = 3 + 1
[5,4,1,1]
=> 101000110 => [1,2,4,1,2] => ? = 2 + 1
[5,3,2,1]
=> 100101010 => [1,3,2,2,2] => ? = 2 + 1
[5,2,2,2]
=> 100011100 => [1,4,1,1,3] => ? = 2 + 1
[4,4,1,1,1]
=> 110001110 => [1,1,4,1,1,2] => ? = 2 + 1
[4,3,2,1,1]
=> 101010110 => [1,2,2,2,1,2] => ? = 1 + 1
[4,2,2,2,1]
=> 100111010 => [1,3,1,1,2,2] => ? = 2 + 1
[3,3,2,1,1,1]
=> 110101110 => [1,1,2,2,1,1,2] => ? = 1 + 1
[3,2,2,2,1,1]
=> 101110110 => [1,2,1,1,2,1,2] => ? = 1 + 1
[2,2,2,2,1,1,1]
=> 111101110 => [1,1,1,1,2,1,1,2] => ? = 1 + 1
[7,5]
=> 100100000 => [1,3,6] => ? = 2 + 1
[6,5,1]
=> 101000010 => [1,2,5,2] => ? = 2 + 1
[6,4,2]
=> 100100100 => [1,3,3,3] => ? = 1 + 1
[6,3,3]
=> 100011000 => [1,4,1,4] => ? = 1 + 1
[5,5,1,1]
=> 110000110 => [1,1,5,1,2] => ? = 2 + 1
[5,4,2,1]
=> 101001010 => [1,2,3,2,2] => ? = 2 + 1
[5,3,3,1]
=> 100110010 => [1,3,1,3,2] => ? = 2 + 1
[5,3,2,2]
=> 100101100 => [1,3,2,1,3] => ? = 2 + 1
[4,4,2,1,1]
=> 110010110 => [1,1,3,2,1,2] => ? = 2 + 1
[4,3,3,1,1]
=> 101100110 => [1,2,1,3,1,2] => ? = 2 + 1
[4,3,2,2,1]
=> 101011010 => [1,2,2,1,2,2] => ? = 1 + 1
[4,2,2,2,2]
=> 100111100 => [1,3,1,1,1,3] => ? = 1 + 1
[3,3,3,1,1,1]
=> 111001110 => [1,1,1,3,1,1,2] => ? = 2 + 1
Description
The number of different parts of an integer composition.
Matching statistic: St000905
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000905: Integer compositions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000905: Integer compositions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1]
=> [[1]]
=> [1] => 1
[2]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
[1,1]
=> [2]
=> [[1,2]]
=> [2] => 1
[3]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[2,1]
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[1,1,1]
=> [3]
=> [[1,2,3]]
=> [3] => 1
[4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 2
[2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 1
[2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> [4] => 1
[5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 2
[3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 2
[3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 2
[2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 1
[2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 1
[6]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[5,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 2
[4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 2
[3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 1
[3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
[3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 2
[2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 1
[2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => 1
[2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
[1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> [6] => 1
[7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => 1
[6,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => 2
[5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => 2
[5,1,1]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => 2
[4,3]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => 2
[4,2,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => 2
[4,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => 2
[3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => 2
[3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => 2
[3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => 1
[3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => 2
[2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => 1
[2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => 1
[2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => 1
[1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [7] => 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,2] => 2
[6,2]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [1,1,1,1,2,2] => 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,3] => 2
[5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [1,1,2,2,2] => 2
[5,2,1]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [1,1,1,2,3] => 2
[7,3]
=> [2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4],[5],[7],[9]]
=> [1,1,1,1,2,2,2] => ? = 2
[6,4]
=> [2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> [1,1,2,2,2,2] => ? = 2
[6,3,1]
=> [3,2,2,1,1,1]
=> [[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> [1,1,1,2,2,3] => ? = 3
[6,2,2]
=> [3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> [1,1,1,1,3,3] => ? = 2
[6,2,1,1]
=> [4,2,1,1,1,1]
=> [[1,6,9,10],[2,8],[3],[4],[5],[7]]
=> [1,1,1,1,2,4] => ? = 2
[5,5]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 1
[5,4,1]
=> [3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [1,2,2,2,3] => ? = 2
[5,3,2]
=> [3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9],[5],[8]]
=> [1,1,2,3,3] => ? = 2
[5,3,1,1]
=> [4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> [1,1,2,2,4] => ? = 2
[5,2,2,1]
=> [4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [1,1,1,3,4] => ? = 2
[4,4,2]
=> [3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> [2,2,3,3] => ? = 1
[4,4,1,1]
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [2,2,2,4] => ? = 2
[4,3,3]
=> [3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> [1,3,3,3] => ? = 2
[4,3,2,1]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [1,2,3,4] => ? = 1
[4,3,1,1,1]
=> [5,2,2,1]
=> [[1,3,8,9,10],[2,5],[4,7],[6]]
=> [1,2,2,5] => ? = 2
[4,2,2,2]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [1,1,4,4] => ? = 1
[4,2,2,1,1]
=> [5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> [1,1,3,5] => ? = 2
[3,3,3,1]
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [3,3,4] => ? = 2
[3,3,2,2]
=> [4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> [2,4,4] => ? = 2
[3,3,2,1,1]
=> [5,3,2]
=> [[1,2,5,9,10],[3,4,8],[6,7]]
=> [2,3,5] => ? = 1
[3,3,1,1,1,1]
=> [6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [2,2,6] => ? = 2
[3,2,2,2,1]
=> [5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [1,4,5] => ? = 1
[3,2,2,1,1,1]
=> [6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [1,3,6] => ? = 1
[2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [5,5] => ? = 1
[2,2,2,2,1,1]
=> [6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [4,6] => ? = 1
[2,2,2,1,1,1,1]
=> [7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [3,7] => ? = 1
[7,4]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? => ? = 2
[6,5]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 2
[6,4,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 3
[6,3,2]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 3
[5,5,1]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 2
[5,4,2]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,3] => ? = 2
[5,4,1,1]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [1,2,2,2,4] => ? = 2
[5,3,3]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [1,1,3,3,3] => ? = 2
[5,3,2,1]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [1,1,2,3,4] => ? = 2
[5,2,2,2]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> [1,1,1,4,4] => ? = 2
[4,4,3]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [2,3,3,3] => ? = 2
[4,4,2,1]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [2,2,3,4] => ? = 2
[4,4,1,1,1]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [2,2,2,5] => ? = 2
[4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [1,3,3,4] => ? = 2
[4,3,2,2]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [1,2,4,4] => ? = 2
[4,3,2,1,1]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [1,2,3,5] => ? = 1
[4,2,2,2,1]
=> [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [1,1,4,5] => ? = 2
[3,3,3,2]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [3,4,4] => ? = 2
[3,3,3,1,1]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [3,3,5] => ? = 2
[3,3,2,2,1]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [2,4,5] => ? = 1
[3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 1
[3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? = 2
[3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 1
[2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 1
Description
The number of different multiplicities of parts of an integer composition.
Matching statistic: St001964
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1,0]
=> ([],1)
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 2 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 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]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[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]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 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]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 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]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 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]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 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]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 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]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[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]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 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]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 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]
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2 - 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]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 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]
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2 - 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]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 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]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 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]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 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]
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[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]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 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]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1 - 1
[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]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 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]
=> ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,15),(3,13),(4,12),(5,3),(5,19),(6,1),(6,17),(7,2),(7,17),(9,11),(10,8),(11,8),(12,9),(13,10),(14,9),(14,18),(15,14),(15,19),(16,12),(16,14),(17,15),(17,16),(18,10),(18,11),(19,13),(19,18)],20)
=> ? = 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]
=> ([(0,6),(0,7),(2,5),(2,16),(3,4),(3,15),(4,9),(5,10),(6,3),(6,14),(7,2),(7,14),(8,1),(9,11),(10,12),(11,8),(12,8),(13,11),(13,12),(14,15),(14,16),(15,9),(15,13),(16,10),(16,13)],17)
=> ? = 2 - 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]
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 2 - 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]
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ? = 2 - 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]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 2 - 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]
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2 - 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]
=> ([(0,6),(0,7),(2,10),(3,9),(4,1),(5,4),(6,3),(6,8),(7,2),(7,8),(8,9),(8,10),(9,11),(10,11),(11,5)],12)
=> ? = 2 - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 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]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 2 - 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]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 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]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2 - 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]
=> ([(0,3),(0,7),(2,9),(3,8),(4,6),(5,4),(6,1),(7,2),(7,8),(8,9),(9,5)],10)
=> ? = 2 - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 - 1
[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]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 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]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1 - 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]
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 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]
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> ? = 2 - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[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]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[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]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1 - 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]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[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]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ([(0,6),(0,7),(2,5),(2,16),(3,4),(3,15),(4,9),(5,10),(6,3),(6,14),(7,2),(7,14),(8,1),(9,11),(10,12),(11,8),(12,8),(13,11),(13,12),(14,15),(14,16),(15,9),(15,13),(16,10),(16,13)],17)
=> ? = 2 - 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]
=> ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,8),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,10),(12,11),(12,13),(13,8),(13,10)],14)
=> ? = 1 - 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ([(0,6),(0,7),(2,9),(3,10),(4,1),(5,3),(5,11),(6,5),(6,8),(7,2),(7,8),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,4)],14)
=> ? = 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]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 2 - 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 2 - 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2 - 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
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