Your data matches 528 different statistics following compositions of up to 3 maps.
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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
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.
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000092: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => 1
[2]
=> [[1,2]]
=> [1,2] => 1
[1,1]
=> [[1],[2]]
=> [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 2
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 3
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 3
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
Description
The number of outer peaks of a permutation. An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$. In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
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
Description
Half the length of a longest factor which is its own reverse-complement of a binary word.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
St000242: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 2 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 3 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1 = 2 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 1 = 2 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 3 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1 = 2 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2 = 3 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 1 = 2 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 3 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1 = 2 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0 = 1 - 1
Description
The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000647: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => 0 = 1 - 1
[2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 2 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 2 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 2 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1 = 2 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1 = 2 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 2 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 3 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1 = 2 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1 = 2 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1 = 2 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1 = 2 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 1 = 2 - 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 3 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1 = 2 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 3 - 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 2 = 3 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1 = 2 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0 = 1 - 1
Description
The number of big descents of a permutation. For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$. The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1]. For the number of small descents, see [[St000214]].
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St001469: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => 0 = 1 - 1
[2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1 = 2 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0 = 1 - 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 2 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1 = 2 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1 = 2 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0 = 1 - 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1 = 2 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1 = 2 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1 = 2 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 3 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1 = 2 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0 = 1 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 1 = 2 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 1 = 2 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 1 = 2 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 1 = 2 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 2 = 3 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 2 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 3 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2 = 3 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 1 = 2 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0 = 1 - 1
Description
The holeyness of a permutation. For $S\subset [n]:=\{1,2,\dots,n\}$ let $\delta(S)$ be the number of elements $m\in S$ such that $m+1\notin S$. For a permutation $\pi$ of $[n]$ the holeyness of $\pi$ is $$\max_{S\subset [n]} (\delta(\pi(S))-\delta(S)).$$
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00155: Standard tableaux promotionStandard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [[1]]
=> 0 = 1 - 1
[2]
=> [[1,2]]
=> [[1,2]]
=> 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [[1],[2]]
=> 0 = 1 - 1
[3]
=> [[1,2,3]]
=> [[1,2,3]]
=> 0 = 1 - 1
[2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1 = 2 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> 0 = 1 - 1
[4]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> 0 = 1 - 1
[3,1]
=> [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 1 = 2 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3],[2],[4]]
=> 1 = 2 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> 0 = 1 - 1
[5]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [[1,3,4],[2,5]]
=> 1 = 2 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,3,4],[2],[5]]
=> 1 = 2 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3],[2],[4],[5]]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> 0 = 1 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> 0 = 1 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1 = 2 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,3,4,5],[2,6]]
=> 1 = 2 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,3,4,5],[2],[6]]
=> 1 = 2 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,3,4],[2,5,6]]
=> 1 = 2 - 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,3,4],[2,6],[5]]
=> 2 = 3 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,3,4],[2],[5],[6]]
=> 1 = 2 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> 2 = 3 - 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3],[2,5],[4],[6]]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3],[2],[4],[5],[6]]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 0 = 1 - 1
Description
The number of natural descents of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00295: Standard tableaux valley compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1]
=> 1
[2]
=> [[1,2]]
=> [2] => [2]
=> 1
[1,1]
=> [[1],[2]]
=> [2] => [2]
=> 1
[3]
=> [[1,2,3]]
=> [3] => [3]
=> 1
[2,1]
=> [[1,3],[2]]
=> [2,1] => [2,1]
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [3] => [3]
=> 1
[4]
=> [[1,2,3,4]]
=> [4] => [4]
=> 1
[3,1]
=> [[1,3,4],[2]]
=> [2,2] => [2,2]
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [3,1] => [3,1]
=> 2
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,1] => [3,1]
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4] => [4]
=> 1
[5]
=> [[1,2,3,4,5]]
=> [5] => [5]
=> 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,3] => [3,2]
=> 2
[3,2]
=> [[1,2,5],[3,4]]
=> [3,2] => [3,2]
=> 2
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2] => [3,2]
=> 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [2,2,1] => [2,2,1]
=> 3
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,1] => [4,1]
=> 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5] => [5]
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [6]
=> 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,4] => [4,2]
=> 2
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,3] => [3,3]
=> 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,3] => [3,3]
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,2] => [4,2]
=> 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [2,2,2] => [2,2,2]
=> 3
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,2] => [4,2]
=> 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [3,2,1] => [3,2,1]
=> 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 3
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,1] => [5,1]
=> 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6] => [6]
=> 1
Description
The length of the partition.
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00294: Standard tableaux peak compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000097: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 1
[2]
=> [[1,2]]
=> [2] => ([],2)
=> 1
[1,1]
=> [[1],[2]]
=> [2] => ([],2)
=> 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [3] => ([],3)
=> 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4] => ([],4)
=> 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6] => ([],6)
=> 1
Description
The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00294: Standard tableaux peak compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000098: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 1
[2]
=> [[1,2]]
=> [2] => ([],2)
=> 1
[1,1]
=> [[1],[2]]
=> [2] => ([],2)
=> 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [3] => ([],3)
=> 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4] => ([],4)
=> 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6] => ([],6)
=> 1
Description
The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
The following 518 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000099The number of valleys of a permutation, including the boundary. St000141The maximum drop size of a permutation. St000172The Grundy number of a graph. St000183The side length of the Durfee square of an integer partition. St000201The number of leaf nodes in a binary tree. St000288The number of ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000783The side length of the largest staircase partition fitting into a partition. St000822The Hadwiger number of the graph. St000875The semilength of the longest Dyck word in the Catalan factorisation of 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. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001322The size of a minimal independent dominating set in a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001432The order dimension of the partition. St001494The Alon-Tarsi number of a graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001716The 1-improper chromatic number of a graph. St001734The lettericity of a graph. St001829The common independence number of a graph. St001924The number of cells in an integer partition whose arm and leg length coincide. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000057The Shynar inversion number of a standard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000204The number of internal nodes of a binary tree. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000317The cycle descent number of a permutation. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000536The pathwidth of a graph. St000547The number of even non-empty partial sums of an integer partition. St000662The staircase size of the code of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001277The degeneracy of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001394The genus of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St000619The number of cyclic descents of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000291The number of descents of a binary word. St000353The number of inner valleys of a permutation. St001388The number of non-attacking neighbors of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000568The hook number of a binary tree. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000497The lcb statistic of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000646The number of big ascents of a permutation. St000691The number of changes of a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001489The maximum of the number of descents and the number of inverse descents. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001727The number of invisible inversions of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000834The number of right outer peaks of a permutation. St001928The number of non-overlapping descents in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000989The number of final rises of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001330The hat guessing number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000308The height of the tree associated to a permutation. St000015The number of peaks of a Dyck path. St000060The greater neighbor of the maximum. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000495The number of inversions of distance at most 2 of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001375The pancake length of a permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001471The magnitude of a Dyck path. St001486The number of corners of the ribbon associated with an integer composition. 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. St001530The depth of a Dyck path. St001569The maximal modular displacement of a permutation. St001729The number of visible descents of a permutation. St001732The number of peaks visible from the left. St001737The number of descents of type 2 in 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. St000365The number of 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. St001960The number of descents of a permutation minus one if its first entry is not one. St000789The number of crossing-similar perfect matchings of a perfect matching. St001220The width of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001557The number of inversions of the second entry 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 $eA$. St000075The orbit size of a standard tableau under promotion. St000706The product of the factorials of the multiplicities of an integer partition. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000939The number of characters of the symmetric group whose value on the partition is positive. St000260The radius of a connected graph. St000485The length of the longest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000702The number of weak deficiencies of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St001062The maximal size of a block of a set partition. St001487The number of inner corners of a skew partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001462The number of factors of a standard tableaux under concatenation. St001864The number of excedances of a signed permutation. St001862The number of crossings of a signed permutation. St001863The number of weak excedances of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. 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 $eA$. St001380The number of monomer-dimer tilings of a Ferrers diagram. St000454The largest eigenvalue of a graph if it is integral. St001435The number of missing boxes in the first row. St001624The breadth of a lattice. St001423The number of distinct cubes in a binary word. St001438The number of missing boxes of a skew partition. St001811The Castelnuovo-Mumford regularity of a permutation. St000668The least common multiple of the parts of the partition. St000144The pyramid weight of the Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000439The position of the first down step of a Dyck path. St000459The hook length of the base cell of a partition. St000744The length of the path to the largest entry in a standard Young tableau. St000759The smallest missing part in an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001488The number of corners of a skew partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001733The number of weak left to right maxima of a Dyck path. St001814The number of partitions interlacing the given partition. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. 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. 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. 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. 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. St000527The width of the poset. St000031The number of cycles in the cycle decomposition of a permutation. St001060The distinguishing index of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000534The number of 2-rises of a permutation. St000842The breadth of a permutation. 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. 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. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000708The product of the parts of an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St000679The pruning number of an ordered tree. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001568The smallest positive integer that does not appear twice in the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000058The order of a permutation. 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. St000456The monochromatic index of a connected graph. St001162The minimum jump of a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001344The neighbouring number of a permutation. St001360The number of covering relations in Young's lattice below a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. 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. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000871The number of very big ascents of a permutation. St000993The multiplicity of the largest part of an integer partition. St001490The number of connected components of a skew partition. 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. 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. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001875The number of simple modules with projective dimension at most 1. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000234The number of global ascents of a permutation. St000253The crossing number of a set partition. St000264The girth of a graph, which is not a tree. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000486The number of cycles of length at least 3 of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000563The number of overlapping pairs of blocks of a set partition. St000567The sum of the products of all pairs of parts. St000570The Edelman-Greene number of a permutation. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000633The size of the automorphism group of a poset. St000663The number of right floats of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents of a permutation. St000886The number of permutations with the same antidiagonal sums. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001399The distinguishing number of a poset. St001413Half the length of the longest even length palindromic prefix of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001470The cyclic holeyness of a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St000039The number of crossings of a permutation. St000084The number of subtrees. St000105The number of blocks in the set partition. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000251The number of nonsingleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000314The number of left-to-right-maxima of a permutation. St000328The maximum number of child nodes in a tree. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000462The major index minus the number of excedences of a permutation. St000487The length of the shortest cycle of a permutation. St000496The rcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000502The number of successions of a set partitions. St000504The cardinality of the first block of a set partition. St000516The number of stretching pairs of a permutation. St000542The number of left-to-right-minima of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000850The number of 1/2-balanced pairs in a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000943The number of spots the most unlucky car had to go further in a parking function. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001114The number of odd descents of a permutation. St001130The number of two successive successions in a permutation. St001151The number of blocks with odd minimum. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St000638The number of up-down runs of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St001424The number of distinct squares in a binary word. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000632The jump number of the poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000907The number of maximal antichains of minimal length in a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000717The number of ordinal summands of a poset. St000284The Plancherel distribution on integer partitions. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001128The exponens consonantiae of a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000102The charge of a semistandard tableau. St000718The largest Laplacian eigenvalue of a graph if it is integral.