Your data matches 64 different statistics following compositions of up to 3 maps.
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St000290: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 0
10 => 1
11 => 0
000 => 0
001 => 0
010 => 2
011 => 0
100 => 1
101 => 1
110 => 2
111 => 0
0000 => 0
0001 => 0
0010 => 3
0011 => 0
0100 => 2
0101 => 2
0110 => 3
0111 => 0
1000 => 1
1001 => 1
1010 => 4
1011 => 1
1100 => 2
1101 => 2
1110 => 3
1111 => 0
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
St000293: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 0
10 => 1
11 => 0
000 => 0
001 => 0
010 => 1
011 => 0
100 => 2
101 => 1
110 => 2
111 => 0
0000 => 0
0001 => 0
0010 => 1
0011 => 0
0100 => 2
0101 => 1
0110 => 2
0111 => 0
1000 => 3
1001 => 2
1010 => 3
1011 => 1
1100 => 4
1101 => 2
1110 => 3
1111 => 0
Description
The number of inversions of a binary word.
Matching statistic: St001438
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
St001438: Skew partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> 0
1 => [1,1] => [[1,1],[]]
=> 0
00 => [3] => [[3],[]]
=> 0
01 => [2,1] => [[2,2],[1]]
=> 1
10 => [1,2] => [[2,1],[]]
=> 0
11 => [1,1,1] => [[1,1,1],[]]
=> 0
000 => [4] => [[4],[]]
=> 0
001 => [3,1] => [[3,3],[2]]
=> 2
010 => [2,2] => [[3,2],[1]]
=> 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> 2
100 => [1,3] => [[3,1],[]]
=> 0
101 => [1,2,1] => [[2,2,1],[1]]
=> 1
110 => [1,1,2] => [[2,1,1],[]]
=> 0
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> 0
0000 => [5] => [[5],[]]
=> 0
0001 => [4,1] => [[4,4],[3]]
=> 3
0010 => [3,2] => [[4,3],[2]]
=> 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> 4
0100 => [2,3] => [[4,2],[1]]
=> 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> 3
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 3
1000 => [1,4] => [[4,1],[]]
=> 0
1001 => [1,3,1] => [[3,3,1],[2]]
=> 2
1010 => [1,2,2] => [[3,2,1],[1]]
=> 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 2
1100 => [1,1,3] => [[3,1,1],[]]
=> 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 0
Description
The number of missing boxes of a skew partition.
Matching statistic: St000175
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00182: Skew partitions outer shapeInteger partitions
St000175: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> [2]
=> 0
1 => [1,1] => [[1,1],[]]
=> [1,1]
=> 0
00 => [3] => [[3],[]]
=> [3]
=> 0
01 => [2,1] => [[2,2],[1]]
=> [2,2]
=> 0
10 => [1,2] => [[2,1],[]]
=> [2,1]
=> 1
11 => [1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> 0
000 => [4] => [[4],[]]
=> [4]
=> 0
001 => [3,1] => [[3,3],[2]]
=> [3,3]
=> 0
010 => [2,2] => [[3,2],[1]]
=> [3,2]
=> 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> 0
100 => [1,3] => [[3,1],[]]
=> [3,1]
=> 1
101 => [1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> 2
110 => [1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> 2
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> 0
0000 => [5] => [[5],[]]
=> [5]
=> 0
0001 => [4,1] => [[4,4],[3]]
=> [4,4]
=> 0
0010 => [3,2] => [[4,3],[2]]
=> [4,3]
=> 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> 0
0100 => [2,3] => [[4,2],[1]]
=> [4,2]
=> 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 0
1000 => [1,4] => [[4,1],[]]
=> [4,1]
=> 1
1001 => [1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> 2
1010 => [1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> 3
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> 3
1100 => [1,1,3] => [[3,1,1],[]]
=> [3,1,1]
=> 2
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> 4
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> [2,1,1,1]
=> 3
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 0
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial $$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$ The statistic of the degree of this polynomial. For example, the partition $(3, 2, 1, 1, 1)$ gives $$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$ which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$. This is the same as the number of unordered pairs of different parts, which follows from: $$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000228
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> []
=> 0
1 => [1,1] => [[1,1],[]]
=> []
=> 0
00 => [3] => [[3],[]]
=> []
=> 0
01 => [2,1] => [[2,2],[1]]
=> [1]
=> 1
10 => [1,2] => [[2,1],[]]
=> []
=> 0
11 => [1,1,1] => [[1,1,1],[]]
=> []
=> 0
000 => [4] => [[4],[]]
=> []
=> 0
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
010 => [2,2] => [[3,2],[1]]
=> [1]
=> 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
100 => [1,3] => [[3,1],[]]
=> []
=> 0
101 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
110 => [1,1,2] => [[2,1,1],[]]
=> []
=> 0
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> 0
0000 => [5] => [[5],[]]
=> []
=> 0
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 3
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 4
0100 => [2,3] => [[4,2],[1]]
=> [1]
=> 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 3
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
1000 => [1,4] => [[4,1],[]]
=> []
=> 0
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
1010 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
1100 => [1,1,3] => [[3,1,1],[]]
=> []
=> 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> 0
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St000355: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,1] => 0
1 => [1,1] => [1,0,1,0]
=> [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
Description
The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00229: Dyck paths Delest-ViennotDyck paths
St000369: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
Description
The dinv deficit of a Dyck path. For a Dyck path $D$ of semilength $n$, this is defined as $$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$ In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$. See also [[St000376]] for the bounce deficit.
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St000462: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,1] => 0
1 => [1,1] => [1,0,1,0]
=> [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
Description
The major index minus the number of excedences of a permutation. This occurs in the context of Eulerian polynomials [1].
Matching statistic: St000516
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000516: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,3,1] => 0
1 => [1,1] => [1,0,1,0]
=> [3,1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
Description
The number of stretching pairs of a permutation. This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Matching statistic: St000589
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000589: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> {{1,2}}
=> 0
1 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 0
00 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
The following 54 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000747A variant of the major index of a set partition. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001175The size of a partition minus the hook length of the base cell. St001377The major index minus the number of inversions of a permutation. St001485The modular major index of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000455The second largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. 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. St001100The 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 trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by 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. St001651The Frankl number of a lattice.