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Your data matches 251 different statistics following compositions of up to 3 maps.
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Matching statistic: St000713
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Mp00311: Plane partitions —to partition⟶ Integer partitions
St000713: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000713: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> 0 = 1 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 0 = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> 0 = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> 0 = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 0 = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> 0 = 1 - 1
[[3],[3],[1]]
=> [3,3,1]
=> 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> 0 = 1 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> 0 = 1 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 0 = 1 - 1
Description
The dimension of the irreducible representation of Sp(4) labelled by an integer partition.
Consider the symplectic group $Sp(2n)$. Then the integer partition $(\mu_1,\dots,\mu_k)$ of length at most $n$ corresponds to the weight vector $(\mu_1-\mu_2,\dots,\mu_{k-2}-\mu_{k-1},\mu_n,0,\dots,0)$.
For example, the integer partition $(2)$ labels the symmetric square of the vector representation, whereas the integer partition $(1,1)$ labels the second fundamental representation.
Matching statistic: St000714
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> 0 = 1 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 0 = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> 0 = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> 0 = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 0 = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> 0 = 1 - 1
[[3],[3],[1]]
=> [3,3,1]
=> 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> 0 = 1 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> 0 = 1 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 0 = 1 - 1
Description
The number of semistandard Young tableau of given shape, with entries at most 2.
This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St000326
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> 1110 => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 11110 => 1
[[2],[1],[1]]
=> [2,1,1]
=> 10110 => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> 10110 => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 111110 => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 1
[[2],[2],[1]]
=> [2,2,1]
=> 11010 => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 11010 => 1
[[3],[1],[1]]
=> [3,1,1]
=> 100110 => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> 100110 => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 100110 => 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 1111110 => 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 110110 => 1
[[2],[2],[2]]
=> [2,2,2]
=> 11100 => 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> 110110 => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> 11100 => 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => 1
[[3],[2],[1]]
=> [3,2,1]
=> 101010 => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> 101010 => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> 101010 => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> 101010 => 1
[[4],[1],[1]]
=> [4,1,1]
=> 1000110 => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> 1000110 => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> 1000110 => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> 1000110 => 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> 1000110 => 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> 11111110 => 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 10111110 => 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> 1101110 => 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> 111010 => 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 10111110 => 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> 1101110 => 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> 111010 => 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 1
[[3],[2],[2]]
=> [3,2,2]
=> 101100 => 1
[[3],[3],[1]]
=> [3,3,1]
=> 110010 => 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 1
[[2,1],[2],[2]]
=> [3,2,2]
=> 101100 => 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> 101100 => 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> 110010 => 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000296
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> 1110 => 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 11110 => 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> 10110 => 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> 10110 => 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 111110 => 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> 11010 => 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 11010 => 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> 100110 => 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> 100110 => 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 100110 => 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 1111110 => 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 110110 => 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> 11100 => 0 = 1 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> 110110 => 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> 11100 => 0 = 1 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> 101010 => 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> 101010 => 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> 101010 => 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> 101010 => 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> 1000110 => 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> 1000110 => 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> 1000110 => 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> 1000110 => 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> 1000110 => 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> 11111110 => 0 = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 10111110 => 0 = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> 1101110 => 0 = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> 111010 => 0 = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 10111110 => 0 = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> 1101110 => 0 = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> 111010 => 0 = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => 0 = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> 101100 => 0 = 1 - 1
[[3],[3],[1]]
=> [3,3,1]
=> 110010 => 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> 101100 => 0 = 1 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> 101100 => 0 = 1 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> 110010 => 0 = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => 0 = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 0 = 1 - 1
Description
The length of the symmetric border of a binary word.
The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix.
The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
Matching statistic: St000629
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> 1110 => 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 11110 => 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> 10110 => 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> 10110 => 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 111110 => 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> 11010 => 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 11010 => 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> 100110 => 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> 100110 => 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 100110 => 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 1111110 => 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 110110 => 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> 11100 => 0 = 1 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> 110110 => 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> 11100 => 0 = 1 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> 101010 => 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> 101010 => 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> 101010 => 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> 101010 => 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> 1000110 => 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> 1000110 => 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> 1000110 => 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> 1000110 => 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> 1000110 => 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> 11111110 => 0 = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 10111110 => 0 = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> 1101110 => 0 = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> 111010 => 0 = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 10111110 => 0 = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> 1101110 => 0 = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> 111010 => 0 = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => 0 = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> 101100 => 0 = 1 - 1
[[3],[3],[1]]
=> [3,3,1]
=> 110010 => 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> 101100 => 0 = 1 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> 101100 => 0 = 1 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> 110010 => 0 = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => 0 = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 1010110 => 0 = 1 - 1
Description
The defect of a binary word.
The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
Matching statistic: St000929
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00177: Plane partitions —transpose⟶ Plane partitions
Mp00311: Plane partitions —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00311: Plane partitions —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [[1,1,1]]
=> [3]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [[1,1,1,1]]
=> [4]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [[2,1,1]]
=> [4]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [[1,1,1],[1]]
=> [3,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [[1,1,1,1,1]]
=> [5]
=> 0 = 1 - 1
[[2],[1],[1],[1]]
=> [[2,1,1,1]]
=> [5]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [[2,2,1]]
=> [5]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [[1,1,1,1],[1]]
=> [4,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [[1,1,1],[1,1]]
=> [3,2]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [[3,1,1]]
=> [5]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [[2,1,1],[1]]
=> [4,1]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [[1,1,1],[1],[1]]
=> [3,1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [[1,1,1,1,1,1]]
=> [6]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [[2,1,1,1,1]]
=> [6]
=> 0 = 1 - 1
[[2],[2],[1],[1]]
=> [[2,2,1,1]]
=> [6]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [[2,2,2]]
=> [6]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1]]
=> [[1,1,1,1,1],[1]]
=> [5,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [[1,1,1,1],[1,1]]
=> [4,2]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [[1,1,1],[1,1,1]]
=> [3,3]
=> 0 = 1 - 1
[[3],[1],[1],[1]]
=> [[3,1,1,1]]
=> [6]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [[3,2,1]]
=> [6]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [[2,1,1,1],[1]]
=> [5,1]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [[2,2,1],[1]]
=> [5,1]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [[2,1,1],[1,1]]
=> [4,2]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [[1,1,1,1],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [[1,1,1],[1,1],[1]]
=> [3,2,1]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [[4,1,1]]
=> [6]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [[3,1,1],[1]]
=> [5,1]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [[2,1,1],[2]]
=> [4,2]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [[2,1,1],[1],[1]]
=> [4,1,1]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [[1,1,1,1,1,1,1]]
=> [7]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [[2,1,1,1,1,1]]
=> [7]
=> 0 = 1 - 1
[[2],[2],[1],[1],[1]]
=> [[2,2,1,1,1]]
=> [7]
=> 0 = 1 - 1
[[2],[2],[2],[1]]
=> [[2,2,2,1]]
=> [7]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [[1,1,1,1,1,1],[1]]
=> [6,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [[1,1,1,1,1],[1,1]]
=> [5,2]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [[1,1,1,1],[1,1,1]]
=> [4,3]
=> 0 = 1 - 1
[[3],[1],[1],[1],[1]]
=> [[3,1,1,1,1]]
=> [7]
=> 0 = 1 - 1
[[3],[2],[1],[1]]
=> [[3,2,1,1]]
=> [7]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [[3,2,2]]
=> [7]
=> 0 = 1 - 1
[[3],[3],[1]]
=> [[3,3,1]]
=> [7]
=> 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [[2,1,1,1,1],[1]]
=> [6,1]
=> 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [[2,2,1,1],[1]]
=> [6,1]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [[2,2,2],[1]]
=> [6,1]
=> 0 = 1 - 1
[[2,1],[1,1],[1],[1]]
=> [[2,1,1,1],[1,1]]
=> [5,2]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [[2,1,1],[1,1,1]]
=> [4,3]
=> 0 = 1 - 1
[[2,1],[2,1],[1]]
=> [[2,2,1],[1,1]]
=> [5,2]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [[1,1,1,1],[1,1],[1]]
=> [4,2,1]
=> 0 = 1 - 1
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001696
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001696: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001696: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0 = 1 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 0 = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 0 = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 0 = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> 0 = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 0 = 1 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 0 = 1 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 0 = 1 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 0 = 1 - 1
Description
The natural major index 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.
The natural major index of a tableau with natural descent set $D$ is then $\sum_{d\in D} d$.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 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]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 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]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[[3],[3],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000068
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 1
[[3],[3],[1]]
=> [3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 1
Description
The number of minimal elements in a poset.
Matching statistic: St000115
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000115: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000115: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[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
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[2,1,1,1,1,1,0],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[2,2,1,1,1,0,0],[2,1,1,1,1,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[2,1,1,1,1,1,0],[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]]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[2,2,1,1,1,0,0],[2,1,1,1,1,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[2,2,2,1,0,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[3,1,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 1
[[3],[3],[1]]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[3,3,1,0,0,0,0],[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[3,1,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[3,2,2,0,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[3,3,1,0,0,0,0],[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[3,1,1,1,1,0,0],[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[3,2,1,1,0,0,0],[2,2,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> 1
Description
The single entry in the last row.
The following 241 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000382The first part of an integer composition. St000655The length of the minimal rise of a Dyck path. St000742The number of big ascents of a permutation after prepending zero. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000781The number of proper colouring schemes of a Ferrers diagram. St000876The number of factors in the Catalan decomposition of a binary word. St000913The number of ways to refine the partition into singletons. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000042The number of crossings of a perfect matching. St000052The number of valleys of a Dyck path not on the x-axis. St000057The Shynar inversion number of a standard tableau. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000297The number of leading ones in a binary word. St000356The number of occurrences of the pattern 13-2. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000397The Strahler number of a rooted tree. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. 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. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000731The number of double exceedences of a permutation. St000733The row containing the largest entry of a standard tableau. St000842The breadth of a permutation. St000871The number of very big ascents of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000920The logarithmic height of a Dyck path. St000974The length of the trunk of an ordered tree. St001047The maximal number of arcs crossing a given arc of a perfect matching. 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. St001394The genus of a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000047The number of standard immaculate tableaux of a given shape. St000659The number of rises of length at least 2 of a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St000769The major index of a composition regarded as a word. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000234The number of global ascents of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000678The number of up steps after the last double rise of a Dyck path. St000701The protection number of a binary tree. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St000255The number of reduced Kogan faces with the permutation as type. St000078The number of alternating sign matrices whose left key is the permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000358The number of occurrences of the pattern 31-2. 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. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 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. 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. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000056The decomposition (or block) number of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000542The number of left-to-right-minima of a permutation. St000674The number of hills of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. 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 $eA$. St001204Call 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 special CNakayama algebra. 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 $eA$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St000764The number of strong records in an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000761The number of ascents in an integer composition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001732The number of peaks visible from the left. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St000253The crossing number of a set partition. St000657The smallest part of an integer composition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001665The number of pure excedances of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000989The number of final rises of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000007The number of saliances of the permutation. St000214The number of adjacencies of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000124The cardinality of the preimage of the Simion-Schmidt map. St000990The first ascent of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000627The exponent of a binary word. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000666The number of right tethers of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000788The number of nesting-similar perfect matchings of a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. 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 $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000058The order of a permutation. St000091The descent variation of a composition. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. 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. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St000355The number of occurrences of the pattern 21-3. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000485The length of the longest cycle of a permutation. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset.
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