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St000533: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 1
[3,1]
=> 2
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 1
[4,1]
=> 2
[3,2]
=> 2
[3,1,1]
=> 3
[2,2,1]
=> 2
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 1
[5,1]
=> 2
[4,2]
=> 2
[4,1,1]
=> 3
[3,3]
=> 2
[3,2,1]
=> 3
[3,1,1,1]
=> 3
[2,2,2]
=> 2
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
[7]
=> 1
[6,1]
=> 2
[5,2]
=> 2
[5,1,1]
=> 3
[4,3]
=> 2
[4,2,1]
=> 3
[4,1,1,1]
=> 4
[3,3,1]
=> 3
[3,2,2]
=> 3
[3,2,1,1]
=> 3
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 1
[8]
=> 1
[7,1]
=> 2
[6,2]
=> 2
[5,3]
=> 2
[5,2,1]
=> 3
[4,4]
=> 2
Description
The minimum of the number of parts and the size of the first part of an integer partition. This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
Matching statistic: St000183
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00189: Skew partitions rotateSkew partitions
Mp00182: Skew partitions outer shapeInteger partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1],[]]
=> [[1],[]]
=> [1]
=> 1
[2]
=> [[2],[]]
=> [[2],[]]
=> [2]
=> 1
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> 1
[3]
=> [[3],[]]
=> [[3],[]]
=> [3]
=> 1
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> 1
[4]
=> [[4],[]]
=> [[4],[]]
=> [4]
=> 1
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> 2
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 1
[5]
=> [[5],[]]
=> [[5],[]]
=> [5]
=> 1
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> 2
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> 2
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 3
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 2
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 1
[6]
=> [[6],[]]
=> [[6],[]]
=> [6]
=> 1
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [5,5]
=> 2
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [4,4]
=> 2
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> 3
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> [3,3]
=> 2
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 3
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> 3
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> [2,2,2]
=> 2
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 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
[7]
=> [[7],[]]
=> [[7],[]]
=> [7]
=> 1
[6,1]
=> [[6,1],[]]
=> [[6,6],[5]]
=> [6,6]
=> 2
[5,2]
=> [[5,2],[]]
=> [[5,5],[3]]
=> [5,5]
=> 2
[5,1,1]
=> [[5,1,1],[]]
=> [[5,5,5],[4,4]]
=> [5,5,5]
=> 3
[4,3]
=> [[4,3],[]]
=> [[4,4],[1]]
=> [4,4]
=> 2
[4,2,1]
=> [[4,2,1],[]]
=> [[4,4,4],[3,2]]
=> [4,4,4]
=> 3
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> 4
[3,3,1]
=> [[3,3,1],[]]
=> [[3,3,3],[2]]
=> [3,3,3]
=> 3
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [3,3,3]
=> 3
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> 3
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3]
=> 3
[2,2,2,1]
=> [[2,2,2,1],[]]
=> [[2,2,2,2],[1]]
=> [2,2,2,2]
=> 2
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [2,2,2,2,2]
=> 2
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2]
=> 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
[8]
=> [[8],[]]
=> [[8],[]]
=> [8]
=> 1
[7,1]
=> [[7,1],[]]
=> [[7,7],[6]]
=> [7,7]
=> 2
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [6,6]
=> 2
[5,3]
=> [[5,3],[]]
=> [[5,5],[2]]
=> [5,5]
=> 2
[5,2,1]
=> [[5,2,1],[]]
=> [[5,5,5],[4,3]]
=> [5,5,5]
=> 3
[4,4]
=> [[4,4],[]]
=> [[4,4],[]]
=> [4,4]
=> 2
Description
The side length of the Durfee square of an integer partition. Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$. This is also known as the Frobenius rank.
Matching statistic: St001924
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00189: Skew partitions rotateSkew partitions
Mp00182: Skew partitions outer shapeInteger partitions
St001924: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1],[]]
=> [[1],[]]
=> [1]
=> 1
[2]
=> [[2],[]]
=> [[2],[]]
=> [2]
=> 1
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> 1
[3]
=> [[3],[]]
=> [[3],[]]
=> [3]
=> 1
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> 1
[4]
=> [[4],[]]
=> [[4],[]]
=> [4]
=> 1
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> 2
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 1
[5]
=> [[5],[]]
=> [[5],[]]
=> [5]
=> 1
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> 2
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> 2
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 3
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 2
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 1
[6]
=> [[6],[]]
=> [[6],[]]
=> [6]
=> 1
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [5,5]
=> 2
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [4,4]
=> 2
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> 3
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> [3,3]
=> 2
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 3
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> 3
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> [2,2,2]
=> 2
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 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
[7]
=> [[7],[]]
=> [[7],[]]
=> [7]
=> 1
[6,1]
=> [[6,1],[]]
=> [[6,6],[5]]
=> [6,6]
=> 2
[5,2]
=> [[5,2],[]]
=> [[5,5],[3]]
=> [5,5]
=> 2
[5,1,1]
=> [[5,1,1],[]]
=> [[5,5,5],[4,4]]
=> [5,5,5]
=> 3
[4,3]
=> [[4,3],[]]
=> [[4,4],[1]]
=> [4,4]
=> 2
[4,2,1]
=> [[4,2,1],[]]
=> [[4,4,4],[3,2]]
=> [4,4,4]
=> 3
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> 4
[3,3,1]
=> [[3,3,1],[]]
=> [[3,3,3],[2]]
=> [3,3,3]
=> 3
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [3,3,3]
=> 3
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> 3
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3]
=> 3
[2,2,2,1]
=> [[2,2,2,1],[]]
=> [[2,2,2,2],[1]]
=> [2,2,2,2]
=> 2
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [2,2,2,2,2]
=> 2
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2]
=> 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
[8]
=> [[8],[]]
=> [[8],[]]
=> [8]
=> 1
[7,1]
=> [[7,1],[]]
=> [[7,7],[6]]
=> [7,7]
=> 2
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [6,6]
=> 2
[5,3]
=> [[5,3],[]]
=> [[5,5],[2]]
=> [5,5]
=> 2
[5,2,1]
=> [[5,2,1],[]]
=> [[5,5,5],[4,3]]
=> [5,5,5]
=> 3
[4,4]
=> [[4,4],[]]
=> [[4,4],[]]
=> [4,4]
=> 2
Description
The number of cells in an integer partition whose arm and leg length coincide.
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
Mp00104: Binary words reverseBinary words
St000875: Binary words ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 10 => 1
[2]
=> 100 => 001 => 100 => 1
[1,1]
=> 110 => 011 => 110 => 1
[3]
=> 1000 => 0001 => 1000 => 1
[2,1]
=> 1010 => 0011 => 1100 => 2
[1,1,1]
=> 1110 => 0111 => 1110 => 1
[4]
=> 10000 => 00001 => 10000 => 1
[3,1]
=> 10010 => 00011 => 11000 => 2
[2,2]
=> 1100 => 0011 => 1100 => 2
[2,1,1]
=> 10110 => 00111 => 11100 => 2
[1,1,1,1]
=> 11110 => 01111 => 11110 => 1
[5]
=> 100000 => 000001 => 100000 => 1
[4,1]
=> 100010 => 000011 => 110000 => 2
[3,2]
=> 10100 => 00011 => 11000 => 2
[3,1,1]
=> 100110 => 000111 => 111000 => 3
[2,2,1]
=> 11010 => 00111 => 11100 => 2
[2,1,1,1]
=> 101110 => 001111 => 111100 => 2
[1,1,1,1,1]
=> 111110 => 011111 => 111110 => 1
[6]
=> 1000000 => 0000001 => 1000000 => 1
[5,1]
=> 1000010 => 0000011 => 1100000 => 2
[4,2]
=> 100100 => 000011 => 110000 => 2
[4,1,1]
=> 1000110 => 0000111 => 1110000 => 3
[3,3]
=> 11000 => 00011 => 11000 => 2
[3,2,1]
=> 101010 => 001011 => 110100 => 3
[3,1,1,1]
=> 1001110 => 0001111 => 1111000 => 3
[2,2,2]
=> 11100 => 00111 => 11100 => 2
[2,2,1,1]
=> 110110 => 001111 => 111100 => 2
[2,1,1,1,1]
=> 1011110 => 0011111 => 1111100 => 2
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 1111110 => 1
[7]
=> 10000000 => 00000001 => 10000000 => 1
[6,1]
=> 10000010 => 00000011 => 11000000 => 2
[5,2]
=> 1000100 => 0000011 => 1100000 => 2
[5,1,1]
=> 10000110 => 00000111 => 11100000 => 3
[4,3]
=> 101000 => 000011 => 110000 => 2
[4,2,1]
=> 1001010 => 0001011 => 1101000 => 3
[4,1,1,1]
=> 10001110 => 00001111 => 11110000 => 4
[3,3,1]
=> 110010 => 000111 => 111000 => 3
[3,2,2]
=> 101100 => 000111 => 111000 => 3
[3,2,1,1]
=> 1010110 => 0010111 => 1110100 => 3
[3,1,1,1,1]
=> 10011110 => 00011111 => 11111000 => 3
[2,2,2,1]
=> 111010 => 001111 => 111100 => 2
[2,2,1,1,1]
=> 1101110 => 0011111 => 1111100 => 2
[2,1,1,1,1,1]
=> 10111110 => 00111111 => 11111100 => 2
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 11111110 => 1
[8]
=> 100000000 => 000000001 => 100000000 => 1
[7,1]
=> 100000010 => 000000011 => 110000000 => 2
[6,2]
=> 10000100 => 00000011 => 11000000 => 2
[5,3]
=> 1001000 => 0000011 => 1100000 => 2
[5,2,1]
=> 10001010 => 00001011 => 11010000 => 3
[4,4]
=> 110000 => 000011 => 110000 => 2
[8,1]
=> 1000000010 => 0000000011 => 1100000000 => ? = 2
[2,1,1,1,1,1,1,1]
=> 1011111110 => 0011111111 => 1111111100 => ? = 2
[8,2]
=> 1000000100 => 0000000011 => 1100000000 => ? = 2
[2,2,1,1,1,1,1,1]
=> 1101111110 => 0011111111 => 1111111100 => ? = 2
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St000783
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00189: Skew partitions rotateSkew partitions
Mp00182: Skew partitions outer shapeInteger partitions
St000783: Integer partitions ⟶ ℤResult quality: 93% values known / values provided: 93%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1],[]]
=> [[1],[]]
=> [1]
=> 1
[2]
=> [[2],[]]
=> [[2],[]]
=> [2]
=> 1
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> 1
[3]
=> [[3],[]]
=> [[3],[]]
=> [3]
=> 1
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> 1
[4]
=> [[4],[]]
=> [[4],[]]
=> [4]
=> 1
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> 2
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 1
[5]
=> [[5],[]]
=> [[5],[]]
=> [5]
=> 1
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> 2
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> 2
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 3
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 2
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 1
[6]
=> [[6],[]]
=> [[6],[]]
=> [6]
=> 1
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [5,5]
=> 2
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [4,4]
=> 2
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> 3
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> [3,3]
=> 2
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 3
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> 3
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> [2,2,2]
=> 2
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 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
[7]
=> [[7],[]]
=> [[7],[]]
=> [7]
=> 1
[6,1]
=> [[6,1],[]]
=> [[6,6],[5]]
=> [6,6]
=> 2
[5,2]
=> [[5,2],[]]
=> [[5,5],[3]]
=> [5,5]
=> 2
[5,1,1]
=> [[5,1,1],[]]
=> [[5,5,5],[4,4]]
=> [5,5,5]
=> 3
[4,3]
=> [[4,3],[]]
=> [[4,4],[1]]
=> [4,4]
=> 2
[4,2,1]
=> [[4,2,1],[]]
=> [[4,4,4],[3,2]]
=> [4,4,4]
=> 3
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> 4
[3,3,1]
=> [[3,3,1],[]]
=> [[3,3,3],[2]]
=> [3,3,3]
=> 3
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [3,3,3]
=> 3
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> 3
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3]
=> 3
[2,2,2,1]
=> [[2,2,2,1],[]]
=> [[2,2,2,2],[1]]
=> [2,2,2,2]
=> 2
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [2,2,2,2,2]
=> 2
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2]
=> 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
[8]
=> [[8],[]]
=> [[8],[]]
=> [8]
=> 1
[7,1]
=> [[7,1],[]]
=> [[7,7],[6]]
=> [7,7]
=> ? = 2
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [6,6]
=> 2
[5,3]
=> [[5,3],[]]
=> [[5,5],[2]]
=> [5,5]
=> 2
[5,2,1]
=> [[5,2,1],[]]
=> [[5,5,5],[4,3]]
=> [5,5,5]
=> 3
[4,4]
=> [[4,4],[]]
=> [[4,4],[]]
=> [4,4]
=> 2
[4,3,1]
=> [[4,3,1],[]]
=> [[4,4,4],[3,1]]
=> [4,4,4]
=> 3
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? = 2
[8,1]
=> [[8,1],[]]
=> [[8,8],[7]]
=> [8,8]
=> ? = 2
[7,2]
=> [[7,2],[]]
=> [[7,7],[5]]
=> [7,7]
=> ? = 2
[2,2,1,1,1,1,1]
=> [[2,2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? = 2
[2,1,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2,2]
=> ? = 2
[8,2]
=> [[8,2],[]]
=> [[8,8],[6]]
=> [8,8]
=> ? = 2
[7,3]
=> [[7,3],[]]
=> [[7,7],[4]]
=> [7,7]
=> ? = 2
[2,2,2,1,1,1,1]
=> [[2,2,2,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? = 2
[2,2,1,1,1,1,1,1]
=> [[2,2,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2,2]
=> ? = 2
Description
The side length of the largest staircase partition fitting into a partition. For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$. In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram. This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St001432
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00189: Skew partitions rotateSkew partitions
Mp00182: Skew partitions outer shapeInteger partitions
St001432: Integer partitions ⟶ ℤResult quality: 58% values known / values provided: 58%distinct values known / distinct values provided: 75%
Values
[1]
=> [[1],[]]
=> [[1],[]]
=> [1]
=> 1
[2]
=> [[2],[]]
=> [[2],[]]
=> [2]
=> 1
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> 1
[3]
=> [[3],[]]
=> [[3],[]]
=> [3]
=> 1
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> 1
[4]
=> [[4],[]]
=> [[4],[]]
=> [4]
=> 1
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> 2
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 1
[5]
=> [[5],[]]
=> [[5],[]]
=> [5]
=> 1
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> 2
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> 2
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 3
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 2
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 1
[6]
=> [[6],[]]
=> [[6],[]]
=> [6]
=> 1
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [5,5]
=> 2
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [4,4]
=> 2
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> 3
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> [3,3]
=> 2
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 3
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> 3
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> [2,2,2]
=> 2
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 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
[7]
=> [[7],[]]
=> [[7],[]]
=> [7]
=> 1
[6,1]
=> [[6,1],[]]
=> [[6,6],[5]]
=> [6,6]
=> 2
[5,2]
=> [[5,2],[]]
=> [[5,5],[3]]
=> [5,5]
=> 2
[5,1,1]
=> [[5,1,1],[]]
=> [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 3
[4,3]
=> [[4,3],[]]
=> [[4,4],[1]]
=> [4,4]
=> 2
[4,2,1]
=> [[4,2,1],[]]
=> [[4,4,4],[3,2]]
=> [4,4,4]
=> 3
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> ? = 4
[3,3,1]
=> [[3,3,1],[]]
=> [[3,3,3],[2]]
=> [3,3,3]
=> 3
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [3,3,3]
=> 3
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> 3
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3]
=> ? = 3
[2,2,2,1]
=> [[2,2,2,1],[]]
=> [[2,2,2,2],[1]]
=> [2,2,2,2]
=> 2
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [2,2,2,2,2]
=> 2
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2]
=> 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
[8]
=> [[8],[]]
=> [[8],[]]
=> [8]
=> 1
[7,1]
=> [[7,1],[]]
=> [[7,7],[6]]
=> [7,7]
=> ? = 2
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [6,6]
=> 2
[5,3]
=> [[5,3],[]]
=> [[5,5],[2]]
=> [5,5]
=> 2
[5,2,1]
=> [[5,2,1],[]]
=> [[5,5,5],[4,3]]
=> [5,5,5]
=> ? = 3
[4,4]
=> [[4,4],[]]
=> [[4,4],[]]
=> [4,4]
=> 2
[4,3,1]
=> [[4,3,1],[]]
=> [[4,4,4],[3,1]]
=> [4,4,4]
=> 3
[4,2,2]
=> [[4,2,2],[]]
=> [[4,4,4],[2,2]]
=> [4,4,4]
=> 3
[4,2,1,1]
=> [[4,2,1,1],[]]
=> [[4,4,4,4],[3,3,2]]
=> [4,4,4,4]
=> ? = 4
[3,3,2]
=> [[3,3,2],[]]
=> [[3,3,3],[1]]
=> [3,3,3]
=> 3
[3,3,1,1]
=> [[3,3,1,1],[]]
=> [[3,3,3,3],[2,2]]
=> [3,3,3,3]
=> 3
[3,2,2,1]
=> [[3,2,2,1],[]]
=> [[3,3,3,3],[2,1,1]]
=> [3,3,3,3]
=> 3
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,1]]
=> [3,3,3,3,3]
=> ? = 3
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? = 2
[8,1]
=> [[8,1],[]]
=> [[8,8],[7]]
=> [8,8]
=> ? = 2
[7,2]
=> [[7,2],[]]
=> [[7,7],[5]]
=> [7,7]
=> ? = 2
[5,3,1]
=> [[5,3,1],[]]
=> [[5,5,5],[4,2]]
=> [5,5,5]
=> ? = 3
[5,2,2]
=> [[5,2,2],[]]
=> [[5,5,5],[3,3]]
=> [5,5,5]
=> ? = 3
[4,3,1,1]
=> [[4,3,1,1],[]]
=> [[4,4,4,4],[3,3,1]]
=> [4,4,4,4]
=> ? = 4
[4,2,2,1]
=> [[4,2,2,1],[]]
=> [[4,4,4,4],[3,2,2]]
=> [4,4,4,4]
=> ? = 4
[3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2]]
=> [3,3,3,3,3]
=> ? = 3
[3,2,2,1,1]
=> [[3,2,2,1,1],[]]
=> [[3,3,3,3,3],[2,2,1,1]]
=> [3,3,3,3,3]
=> ? = 3
[2,2,1,1,1,1,1]
=> [[2,2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? = 2
[2,1,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2,2]
=> ? = 2
[8,2]
=> [[8,2],[]]
=> [[8,8],[6]]
=> [8,8]
=> ? = 2
[7,3]
=> [[7,3],[]]
=> [[7,7],[4]]
=> [7,7]
=> ? = 2
[5,4,1]
=> [[5,4,1],[]]
=> [[5,5,5],[4,1]]
=> [5,5,5]
=> ? = 3
[5,3,2]
=> [[5,3,2],[]]
=> [[5,5,5],[3,2]]
=> [5,5,5]
=> ? = 3
[4,4,1,1]
=> [[4,4,1,1],[]]
=> [[4,4,4,4],[3,3]]
=> [4,4,4,4]
=> ? = 4
[4,3,2,1]
=> [[4,3,2,1],[]]
=> [[4,4,4,4],[3,2,1]]
=> [4,4,4,4]
=> ? = 4
[4,2,2,2]
=> [[4,2,2,2],[]]
=> [[4,4,4,4],[2,2,2]]
=> [4,4,4,4]
=> ? = 4
[3,3,2,1,1]
=> [[3,3,2,1,1],[]]
=> [[3,3,3,3,3],[2,2,1]]
=> [3,3,3,3,3]
=> ? = 3
[3,2,2,2,1]
=> [[3,2,2,2,1],[]]
=> [[3,3,3,3,3],[2,1,1,1]]
=> [3,3,3,3,3]
=> ? = 3
[2,2,2,1,1,1,1]
=> [[2,2,2,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> ? = 2
[2,2,1,1,1,1,1,1]
=> [[2,2,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2,2]
=> ? = 2
[5,5,1]
=> [[5,5,1],[]]
=> [[5,5,5],[4]]
=> [5,5,5]
=> ? = 3
[5,4,2]
=> [[5,4,2],[]]
=> [[5,5,5],[3,1]]
=> [5,5,5]
=> ? = 3
[5,3,3]
=> [[5,3,3],[]]
=> [[5,5,5],[2,2]]
=> [5,5,5]
=> ? = 3
[4,4,2,1]
=> [[4,4,2,1],[]]
=> [[4,4,4,4],[3,2]]
=> [4,4,4,4]
=> ? = 4
[4,3,3,1]
=> [[4,3,3,1],[]]
=> [[4,4,4,4],[3,1,1]]
=> [4,4,4,4]
=> ? = 4
[4,3,2,2]
=> [[4,3,2,2],[]]
=> [[4,4,4,4],[2,2,1]]
=> [4,4,4,4]
=> ? = 4
[3,3,3,1,1]
=> [[3,3,3,1,1],[]]
=> [[3,3,3,3,3],[2,2]]
=> [3,3,3,3,3]
=> ? = 3
[3,3,2,2,1]
=> [[3,3,2,2,1],[]]
=> [[3,3,3,3,3],[2,1,1]]
=> [3,3,3,3,3]
=> ? = 3
[3,2,2,2,2]
=> [[3,2,2,2,2],[]]
=> [[3,3,3,3,3],[1,1,1,1]]
=> [3,3,3,3,3]
=> ? = 3
[5,5,2]
=> [[5,5,2],[]]
=> [[5,5,5],[3]]
=> [5,5,5]
=> ? = 3
[5,4,3]
=> [[5,4,3],[]]
=> [[5,5,5],[2,1]]
=> [5,5,5]
=> ? = 3
[4,4,3,1]
=> [[4,4,3,1],[]]
=> [[4,4,4,4],[3,1]]
=> [4,4,4,4]
=> ? = 4
[4,4,2,2]
=> [[4,4,2,2],[]]
=> [[4,4,4,4],[2,2]]
=> [4,4,4,4]
=> ? = 4
[4,3,3,2]
=> [[4,3,3,2],[]]
=> [[4,4,4,4],[2,1,1]]
=> [4,4,4,4]
=> ? = 4
[3,3,3,2,1]
=> [[3,3,3,2,1],[]]
=> [[3,3,3,3,3],[2,1]]
=> [3,3,3,3,3]
=> ? = 3
[3,3,2,2,2]
=> [[3,3,2,2,2],[]]
=> [[3,3,3,3,3],[1,1,1]]
=> [3,3,3,3,3]
=> ? = 3
[5,5,3]
=> [[5,5,3],[]]
=> [[5,5,5],[2]]
=> [5,5,5]
=> ? = 3
[5,4,4]
=> [[5,4,4],[]]
=> [[5,5,5],[1,1]]
=> [5,5,5]
=> ? = 3
[4,4,4,1]
=> [[4,4,4,1],[]]
=> [[4,4,4,4],[3]]
=> [4,4,4,4]
=> ? = 4
[4,4,3,2]
=> [[4,4,3,2],[]]
=> [[4,4,4,4],[2,1]]
=> [4,4,4,4]
=> ? = 4
[4,3,3,3]
=> [[4,3,3,3],[]]
=> [[4,4,4,4],[1,1,1]]
=> [4,4,4,4]
=> ? = 4
Description
The order dimension of the partition. Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00185: Skew partitions cell posetPosets
Mp00198: Posets incomparability graphGraphs
St001330: Graphs ⟶ ℤResult quality: 17% values known / values provided: 17%distinct values known / distinct values provided: 75%
Values
[1]
=> [[1],[]]
=> ([],1)
=> ([],1)
=> 1
[2]
=> [[2],[]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 3
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1
[7]
=> [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 1
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 4
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 3
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 3
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> ? = 1
[8]
=> [[8],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 1
[7,1]
=> [[7,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8)
=> ? = 3
[4,4]
=> [[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
[4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 3
[4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(1,6),(1,7),(2,4),(2,5),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> ([(1,2),(1,6),(1,7),(2,4),(2,5),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4
[3,3,2]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> ([(1,4),(1,7),(2,3),(2,7),(3,6),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 3
[3,3,1,1]
=> [[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(1,6),(1,7),(2,4),(2,5),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3
[3,2,2,1]
=> [[3,2,2,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 3
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7)],8)
=> ? = 3
[2,2,2,2]
=> [[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([],8)
=> ? = 1
[9]
=> [[9],[]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([],9)
=> ? = 1
[8,1]
=> [[8,1],[]]
=> ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[7,2]
=> [[7,2],[]]
=> ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(6,1),(7,5),(7,8)],9)
=> ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 2
[6,3]
=> [[6,3],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(4,8),(5,1),(6,4),(6,7),(7,8)],9)
=> ([(1,8),(2,7),(2,8),(3,4),(3,5),(3,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 2
[5,4]
=> [[5,4],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
=> ([(1,8),(2,7),(3,6),(3,7),(4,5),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
[5,3,1]
=> [[5,3,1],[]]
=> ([(0,5),(0,6),(3,4),(3,8),(4,2),(5,3),(5,7),(6,1),(6,7),(7,8)],9)
=> ([(1,7),(1,8),(2,5),(2,6),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 3
[5,2,2]
=> [[5,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9)
=> ([(1,7),(1,8),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 3
[4,4,1]
=> [[4,4,1],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ([(1,8),(2,6),(2,8),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(7,8)],9)
=> ? = 3
[4,3,2]
=> [[4,3,2],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> ([(1,5),(1,8),(2,3),(2,7),(2,8),(3,6),(3,7),(4,6),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[4,3,1,1]
=> [[4,3,1,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> ([(1,6),(1,7),(1,8),(2,3),(2,7),(2,8),(3,5),(3,6),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 4
[4,2,2,1]
=> [[4,2,2,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> ([(1,6),(1,7),(1,8),(2,3),(2,7),(2,8),(3,5),(3,6),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 4
[3,3,3]
=> [[3,3,3],[]]
=> ([(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)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 3
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
Mp00201: Dyck paths RingelPermutations
St001569: Permutations ⟶ ℤResult quality: 11% values known / values provided: 11%distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0]
=> [1,0]
=> [2,1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? = 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ? = 2
[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]
=> [6,1,2,3,4,5] => ? = 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]
=> [2,3,4,5,6,7,1] => ? = 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]
=> [2,3,4,5,7,1,6] => ? = 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ? = 2
[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]
=> [2,3,4,7,1,5,6] => ? = 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 3
[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]
=> [2,3,7,1,4,5,6] => ? = 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 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]
=> [6,3,1,2,4,5] => ? = 2
[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]
=> [2,7,1,3,4,5,6] => ? = 2
[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]
=> [7,1,2,3,4,5,6] => ? = 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]
=> [2,3,4,5,6,7,8,1] => ? = 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]
=> [2,3,4,5,6,8,1,7] => ? = 2
[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]
=> [2,3,4,7,6,1,5] => ? = 2
[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]
=> [2,3,4,5,8,1,6,7] => ? = 3
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ? = 2
[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]
=> [2,3,7,5,1,4,6] => ? = 3
[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]
=> [2,3,4,8,1,5,6,7] => ? = 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 3
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 3
[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]
=> [2,7,4,1,3,5,6] => ? = 3
[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]
=> [2,3,8,1,4,5,6,7] => ? = 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]
=> [6,4,1,2,3,5] => ? = 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]
=> [7,3,1,2,4,5,6] => ? = 2
[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]
=> [2,8,1,3,4,5,6,7] => ? = 2
[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]
=> [8,1,2,3,4,5,6,7] => ? = 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]
=> [2,3,4,5,6,7,8,9,1] => ? = 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]
=> [2,3,4,5,6,7,9,1,8] => ? = 2
[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]
=> [2,3,4,5,8,7,1,6] => ? = 2
[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]
=> [2,3,7,5,6,1,4] => ? = 2
[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]
=> [2,3,4,8,6,1,5,7] => ? = 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ? = 2
[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]
=> [2,7,4,5,1,3,6] => ? = 3
[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]
=> [2,3,7,6,1,4,5] => ? = 3
[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]
=> [2,3,8,5,1,4,6,7] => ? = 4
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ? = 3
[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]
=> [7,3,4,1,2,5,6] => ? = 3
[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]
=> [2,7,5,1,3,4,6] => ? = 3
[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]
=> [2,8,4,1,3,5,6,7] => ? = 3
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 2
[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]
=> [7,4,1,2,3,5,6] => ? = 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]
=> [8,3,1,2,4,5,6,7] => ? = 2
[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]
=> [2,9,1,3,4,5,6,7,8] => ? = 2
[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]
=> [9,1,2,3,4,5,6,7,8] => ? = 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? = 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => ? = 2
[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]
=> [2,3,4,5,6,9,8,1,7] => ? = 2
Description
The maximal modular displacement of a permutation. This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
Matching statistic: St001624
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00185: Skew partitions cell posetPosets
Mp00195: Posets order idealsLattices
St001624: Lattices ⟶ ℤResult quality: 11% values known / values provided: 11%distinct values known / distinct values provided: 50%
Values
[1]
=> [[1],[]]
=> ([],1)
=> ([(0,1)],2)
=> 1
[2]
=> [[2],[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 2
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 3
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 2
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 2
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 2
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? = 3
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ? = 3
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? = 3
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 2
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[7]
=> [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
=> ? = 2
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? = 2
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? = 3
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? = 2
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? = 3
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(0,1),(1,2),(1,3),(2,7),(2,14),(3,6),(3,14),(4,11),(5,12),(6,4),(6,15),(7,5),(7,16),(9,8),(10,8),(11,9),(12,10),(13,9),(13,10),(14,15),(14,16),(15,11),(15,13),(16,12),(16,13)],17)
=> ? = 4
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? = 3
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? = 3
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? = 3
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? = 3
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? = 2
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? = 2
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
=> ? = 2
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1
[8]
=> [[8],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 1
[7,1]
=> [[7,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
=> ? = 2
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
=> ? = 2
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ? = 2
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ?
=> ? = 3
[4,4]
=> [[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 2
[4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? = 3
[4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? = 3
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> ?
=> ? = 4
[3,3,2]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ? = 3
[3,3,1,1]
=> [[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? = 3
[3,2,2,1]
=> [[3,2,2,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? = 3
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ?
=> ? = 3
[2,2,2,2]
=> [[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 2
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ? = 2
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
=> ? = 2
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
=> ? = 2
[1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 1
[9]
=> [[9],[]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 1
[8,1]
=> [[8,1],[]]
=> ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,9),(1,10),(2,11),(3,7),(3,10),(4,6),(4,12),(5,4),(5,14),(6,8),(6,13),(7,5),(7,16),(8,2),(8,15),(9,1),(9,3),(10,16),(12,13),(13,15),(14,12),(15,11),(16,14)],17)
=> ? = 2
[7,2]
=> [[7,2],[]]
=> ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(6,1),(7,5),(7,8)],9)
=> ?
=> ? = 2
Description
The breadth of a lattice. The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001423
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00093: Dyck paths to binary wordBinary words
Mp00268: Binary words zeros to flag zerosBinary words
St001423: Binary words ⟶ ℤResult quality: 11% values known / values provided: 11%distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0]
=> 10 => 01 => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> 1010 => 1001 => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> 1100 => 1011 => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> 101010 => 011001 => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> 101100 => 010001 => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 010011 => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 10011001 => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 10111001 => 1 = 2 - 1
[2,2]
=> [1,1,1,0,0,0]
=> 111000 => 010111 => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 10110001 => 1 = 2 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 10110011 => 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0110011001 => ? = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0100011001 => ? = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 10100001 => 1 = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0100111001 => ? = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 10110111 => 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0100110001 => ? = 2 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0100110011 => ? = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 100110011001 => ? = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 101110011001 => ? = 2 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0101111001 => ? = 2 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => 101100011001 => ? = 3 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 10100111 => 1 = 2 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0100100001 => ? = 3 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => 101100111001 => ? = 3 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 10101111 => 1 = 2 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 0100110111 => ? = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => 101100110001 => ? = 2 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => 101100110011 => ? = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => 01100110011001 => ? = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => 01000110011001 => ? = 2 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 101000011001 => ? = 2 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => 01001110011001 => ? = 3 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0101100001 => ? = 2 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 101011100100 => 101101111001 => ? = 3 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => 01001100011001 => ? = 4 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 0100100111 => ? = 3 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0101000001 => ? = 3 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => 101100100001 => ? = 3 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 10101101010100 => 01001100111001 => ? = 3 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 0100101111 => ? = 2 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 111001010100 => 101100110111 => ? = 2 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 10110101010100 => 01001100110001 => ? = 2 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => 01001100110011 => ? = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1010101010101010 => 1001100110011001 => ? = 1 - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1010101010101100 => 1011100110011001 => ? = 2 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => 01011110011001 => ? = 2 - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => 101001111001 => ? = 2 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => 01001000011001 => ? = 3 - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 0101100111 => ? = 2 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 101110100100 => 101101100001 => ? = 3 - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => 101011111001 => ? = 3 - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 10101110010100 => 01001101111001 => ? = 4 - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 0101000111 => ? = 3 - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 111010010100 => 101100100111 => ? = 3 - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => 101101000001 => ? = 3 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> 10111001010100 => 01001100100001 => ? = 3 - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 0101001111 => ? = 2 - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 111100010100 => 101100101111 => ? = 2 - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 11100101010100 => 01001100110111 => ? = 2 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1011010101010100 => 1011001100110001 => ? = 2 - 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]
=> 1101010101010100 => 1011001100110011 => ? = 1 - 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010101010 => 011001100110011001 => ? = 1 - 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101010101100 => 010001100110011001 => ? = 2 - 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 1010101010111000 => 1010000110011001 => ? = 2 - 1
Description
The number of distinct cubes in a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words $u$ such that $uuu$ is a factor of the word.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001520The number of strict 3-descents. St000527The width of the poset.