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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
[6,1,1]
=> 3
[5,3]
=> 2
[5,2,1]
=> 3
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.
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
St001420: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 1
[2]
=> 100 => 001 => 1
[1,1]
=> 110 => 011 => 1
[3]
=> 1000 => 0001 => 1
[2,1]
=> 1010 => 0011 => 2
[1,1,1]
=> 1110 => 0111 => 1
[4]
=> 10000 => 00001 => 1
[3,1]
=> 10010 => 00011 => 2
[2,2]
=> 1100 => 0011 => 2
[2,1,1]
=> 10110 => 00111 => 2
[1,1,1,1]
=> 11110 => 01111 => 1
[5]
=> 100000 => 000001 => 1
[4,1]
=> 100010 => 000011 => 2
[3,2]
=> 10100 => 00011 => 2
[3,1,1]
=> 100110 => 000111 => 3
[2,2,1]
=> 11010 => 00111 => 2
[2,1,1,1]
=> 101110 => 001111 => 2
[1,1,1,1,1]
=> 111110 => 011111 => 1
[6]
=> 1000000 => 0000001 => 1
[5,1]
=> 1000010 => 0000011 => 2
[4,2]
=> 100100 => 000011 => 2
[4,1,1]
=> 1000110 => 0000111 => 3
[3,3]
=> 11000 => 00011 => 2
[3,2,1]
=> 101010 => 001011 => 3
[3,1,1,1]
=> 1001110 => 0001111 => 3
[2,2,2]
=> 11100 => 00111 => 2
[2,2,1,1]
=> 110110 => 001111 => 2
[2,1,1,1,1]
=> 1011110 => 0011111 => 2
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 1
[7]
=> 10000000 => 00000001 => 1
[6,1]
=> 10000010 => 00000011 => 2
[5,2]
=> 1000100 => 0000011 => 2
[5,1,1]
=> 10000110 => 00000111 => 3
[4,3]
=> 101000 => 000011 => 2
[4,2,1]
=> 1001010 => 0001011 => 3
[4,1,1,1]
=> 10001110 => 00001111 => 4
[3,3,1]
=> 110010 => 000111 => 3
[3,2,2]
=> 101100 => 000111 => 3
[3,2,1,1]
=> 1010110 => 0010111 => 3
[3,1,1,1,1]
=> 10011110 => 00011111 => 3
[2,2,2,1]
=> 111010 => 001111 => 2
[2,2,1,1,1]
=> 1101110 => 0011111 => 2
[2,1,1,1,1,1]
=> 10111110 => 00111111 => 2
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 1
[8]
=> 100000000 => 000000001 => 1
[7,1]
=> 100000010 => 000000011 => 2
[6,2]
=> 10000100 => 00000011 => 2
[6,1,1]
=> 100000110 => 000000111 => 3
[5,3]
=> 1001000 => 0000011 => 2
[5,2,1]
=> 10001010 => 00001011 => 3
Description
Half the length of a longest factor which is its own reverse-complement of a binary word.
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
[6,1,1]
=> [[6,1,1],[]]
=> [[6,6,6],[5,5]]
=> [6,6,6]
=> 3
[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
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: St000875
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
Mp00104: Binary words reverseBinary words
St000875: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%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
[6,1,1]
=> 100000110 => 000000111 => 111000000 => 3
[5,3]
=> 1001000 => 0000011 => 1100000 => 2
[5,2,1]
=> 10001010 => 00001011 => 11010000 => 3
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.
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
Mp00104: Binary words reverseBinary words
St001421: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%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
[6,1,1]
=> 100000110 => 000000111 => 111000000 => 3
[5,3]
=> 1001000 => 0000011 => 1100000 => 2
[5,2,1]
=> 10001010 => 00001011 => 11010000 => 3
Description
Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
Matching statistic: St001036
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
Mp00327: Dyck paths inverse Kreweras complementDyck paths
St001036: Dyck paths ⟶ ℤResult quality: 98% values known / values provided: 98%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 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]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 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]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 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]
=> [1,1,1,1,1,0,0,0,0,0]
=> 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 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]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 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]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 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]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 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]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1 = 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]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 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]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 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]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 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]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 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]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 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]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 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]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 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]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 2 - 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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 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]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> 1 = 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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 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]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 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]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
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: 94% values known / values provided: 94%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
[6,1,1]
=> [[6,1,1],[]]
=> [[6,6,6],[5,5]]
=> [6,6,6]
=> ? ∊ {3,3,4,4}
[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
[5,1,1,1]
=> [[5,1,1,1],[]]
=> [[5,5,5,5],[4,4,4]]
=> [5,5,5,5]
=> ? ∊ {3,3,4,4}
[4,4]
=> [[4,4],[]]
=> [[4,4],[]]
=> [4,4]
=> 2
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,3]]
=> [4,4,4,4,4]
=> ? ∊ {3,3,4,4}
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [3,3,3,3,3,3]
=> ? ∊ {3,3,4,4}
Description
The number of cells in an integer partition whose arm and leg length coincide.
Matching statistic: St000141
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00126: Permutations cactus evacuationPermutations
St000141: Permutations ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,3,4] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,3,4,5] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3,5,4,1] => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1,4] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,3,4,5,6] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [2,3,4,6,5,1] => 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,5,3,4,1] => 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,4,5,2,3] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,3,5,1] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2,5] => 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [3,2,4,5,6,1] => 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,1,3,4,5,6,7] => 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [2,3,4,5,7,6,1] => ? = 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [1,3,4,6,5,2] => 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [2,3,6,4,5,1] => 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,4,2,1] => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,5,2,1,3] => 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,1,3,2] => 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,5,2,1] => 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [4,2,3,5,6,1] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [3,2,4,5,1,6] => 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [3,2,4,5,6,7,1] => 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [2,1,3,4,5,6,7,8] => 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [2,3,4,5,6,8,7,1] => ? ∊ {2,2,2,3}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [1,3,4,5,7,6,2] => ? ∊ {2,2,2,3}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [2,3,4,7,5,6,1] => ? ∊ {2,2,2,3}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,2,4,6,5,3] => 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [3,4,6,5,2,1] => 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [2,6,3,4,5,1] => 4
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [1,2,5,6,3,4] => 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,5,4,3,1] => 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,5,1,4,2] => 3
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => [3,2,4,5,6,1,7] => ? ∊ {2,2,2,3}
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
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: 91% values known / values provided: 91%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]
=> ? ∊ {2,2,3,3,4,4}
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [6,6]
=> 2
[6,1,1]
=> [[6,1,1],[]]
=> [[6,6,6],[5,5]]
=> [6,6,6]
=> ? ∊ {2,2,3,3,4,4}
[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
[5,1,1,1]
=> [[5,1,1,1],[]]
=> [[5,5,5,5],[4,4,4]]
=> [5,5,5,5]
=> ? ∊ {2,2,3,3,4,4}
[4,4]
=> [[4,4],[]]
=> [[4,4],[]]
=> [4,4]
=> 2
[4,3,1]
=> [[4,3,1],[]]
=> [[4,4,4],[3,1]]
=> [4,4,4]
=> 3
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,3]]
=> [4,4,4,4,4]
=> ? ∊ {2,2,3,3,4,4}
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [3,3,3,3,3,3]
=> ? ∊ {2,2,3,3,4,4}
[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]
=> ? ∊ {2,2,3,3,4,4}
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: St000062
Mp00202: Integer partitions first row removalInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St000062: Permutations ⟶ ℤResult quality: 86% values known / values provided: 86%distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> [] => ? = 1
[2]
=> []
=> []
=> [] => ? = 1
[1,1]
=> [1]
=> [1,0]
=> [1] => 1
[3]
=> []
=> []
=> [] => ? = 1
[2,1]
=> [1]
=> [1,0]
=> [1] => 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[4]
=> []
=> []
=> [] => ? = 2
[3,1]
=> [1]
=> [1,0]
=> [1] => 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
[5]
=> []
=> []
=> [] => ? = 3
[4,1]
=> [1]
=> [1,0]
=> [1] => 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[6]
=> []
=> []
=> [] => ? = 3
[5,1]
=> [1]
=> [1,0]
=> [1] => 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 3
[7]
=> []
=> []
=> [] => ? = 3
[6,1]
=> [1]
=> [1,0]
=> [1] => 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 3
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 3
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 4
[8]
=> []
=> []
=> [] => ? ∊ {2,2}
[7,1]
=> [1]
=> [1,0]
=> [1] => 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 3
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 3
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? ∊ {2,2}
Description
The length of the longest increasing subsequence of the permutation.
The following 196 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001432The order dimension of the partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000624The normalized sum of the minimal distances to a greater element. St000619The number of cyclic descents of a permutation. St000711The number of big exceedences of a permutation. St000836The number of descents of distance 2 of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000242The number of indices that are not cyclical small weak excedances. St000099The number of valleys of a permutation, including the boundary. St000392The length of the longest run of ones in a binary word. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000982The length of the longest constant subword. St001322The size of a minimal independent dominating set in a graph. St001829The common independence number of a graph. St000023The number of inner peaks of a permutation. St000647The number of big descents of a permutation. St000837The number of ascents of distance 2 of a permutation. St001388The number of non-attacking neighbors of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001083The number of boxed occurrences of 132 in a permutation. St001394The genus of a permutation. St001469The holeyness of a permutation. St000646The number of big ascents of a permutation. St000308The height of the tree associated to a permutation. St000317The cycle descent number of a permutation. St001569The maximal modular displacement of a permutation. St001729The number of visible descents of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001948The number of augmented double ascents of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000662The staircase size of the code of a permutation. St000075The orbit size of a standard tableau under promotion. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001928The number of non-overlapping descents in a permutation. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000092The number of outer peaks of a permutation. St000353The number of inner valleys of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000822The Hadwiger number of the graph. St001734The lettericity of a graph. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St001557The number of inversions of the second entry of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001812The biclique partition number of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001330The hat guessing number of a graph. St001462The number of factors of a standard tableaux under concatenation. St000744The length of the path to the largest entry in a standard Young tableau. St001060The distinguishing index of a graph. St001863The number of weak excedances of a signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001868The number of alignments of type NE of a signed permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001487The number of inner corners of a skew partition. St001864The number of excedances of a signed permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000454The largest eigenvalue of a graph if it is integral. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001488The number of corners of a skew partition. St001568The smallest positive integer that does not appear twice in the partition. St001624The breadth of a lattice. St001423The number of distinct cubes in a binary word. St001811The Castelnuovo-Mumford regularity of a permutation. St000527The width of the poset. St000291The number of descents of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000390The number of runs of ones in a binary word. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001712The number of natural descents of a standard Young tableau. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000331The number of upper interactions of a Dyck path. St000899The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001820The size of the image of the pop stack sorting operator. St001889The size of the connectivity set of a signed permutation. St001896The number of right descents of a signed permutations. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000015The number of peaks of a Dyck path. St000117The number of centered tunnels of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000648The number of 2-excedences of a permutation. St000664The number of right ropes of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000765The number of weak records in an integer composition. St000942The number of critical left to right maxima of the parking functions. St000991The number of right-to-left minima of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001267The length of the Lyndon factorization of the binary word. St001530The depth of a Dyck path. St001667The maximal size of a pair of weak twins for a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001846The number of elements which do not have a complement in the lattice. St001866The nesting alignments of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000451The length of the longest pattern of the form k 1 2. St000820The number of compositions obtained by rotating the composition. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path). St000031The number of cycles in the cycle decomposition of a permutation. St000264The girth of a graph, which is not a tree. St000534The number of 2-rises of a permutation. St000842The breadth of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000068The number of minimal elements in a poset. St000077The number of boxed and circled entries. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000679The pruning number of an ordered tree. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000090The variation of a composition. St000091The descent variation of a composition. St000217The number of occurrences of the pattern 312 in a permutation. St000233The number of nestings of a set partition. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001705The number of occurrences of the pattern 2413 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000871The number of very big ascents of a permutation. St001490The number of connected components of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000022The number of fixed points of a permutation. St000035The number of left outer peaks of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000546The number of global descents of a permutation. St000731The number of double exceedences of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000862The number of parts of the shifted shape of a permutation.