Your data matches 350 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000297
(load all 14 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 1
[2]
 => 100 => 1
[1,1]
 => 110 => 2
[3]
 => 1000 => 1
[2,1]
 => 1010 => 1
[1,1,1]
 => 1110 => 3
[4]
 => 10000 => 1
[3,1]
 => 10010 => 1
[2,2]
 => 1100 => 2
[2,1,1]
 => 10110 => 1
[1,1,1,1]
 => 11110 => 4
[5]
 => 100000 => 1
[4,1]
 => 100010 => 1
[3,2]
 => 10100 => 1
[3,1,1]
 => 100110 => 1
[2,2,1]
 => 11010 => 2
[2,1,1,1]
 => 101110 => 1
[1,1,1,1,1]
 => 111110 => 5
[6]
 => 1000000 => 1
[5,1]
 => 1000010 => 1
[4,2]
 => 100100 => 1
[4,1,1]
 => 1000110 => 1
[3,3]
 => 11000 => 2
[3,2,1]
 => 101010 => 1
[3,1,1,1]
 => 1001110 => 1
[2,2,2]
 => 11100 => 3
[2,2,1,1]
 => 110110 => 2
[2,1,1,1,1]
 => 1011110 => 1
[1,1,1,1,1,1]
 => 1111110 => 6
[7]
 => 10000000 => 1
[6,1]
 => 10000010 => 1
[5,2]
 => 1000100 => 1
[5,1,1]
 => 10000110 => 1
[4,3]
 => 101000 => 1
[4,2,1]
 => 1001010 => 1
[4,1,1,1]
 => 10001110 => 1
[3,3,1]
 => 110010 => 2
[3,2,2]
 => 101100 => 1
[3,2,1,1]
 => 1010110 => 1
[3,1,1,1,1]
 => 10011110 => 1
[2,2,2,1]
 => 111010 => 3
[2,2,1,1,1]
 => 1101110 => 2
[2,1,1,1,1,1]
 => 10111110 => 1
[1,1,1,1,1,1,1]
 => 11111110 => 7
Description
The number of leading ones in a binary word.
Matching statistic: St000733
(load all 6 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => 1
[2]
 => [[1,2]]
 => 1
[1,1]
 => [[1],[2]]
 => 2
[3]
 => [[1,2,3]]
 => 1
[2,1]
 => [[1,3],[2]]
 => 1
[1,1,1]
 => [[1],[2],[3]]
 => 3
[4]
 => [[1,2,3,4]]
 => 1
[3,1]
 => [[1,3,4],[2]]
 => 1
[2,2]
 => [[1,2],[3,4]]
 => 2
[2,1,1]
 => [[1,4],[2],[3]]
 => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
[5]
 => [[1,2,3,4,5]]
 => 1
[4,1]
 => [[1,3,4,5],[2]]
 => 1
[3,2]
 => [[1,2,5],[3,4]]
 => 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => 2
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
[6]
 => [[1,2,3,4,5,6]]
 => 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => 2
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 3
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 2
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
[7]
 => [[1,2,3,4,5,6,7]]
 => 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 2
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 3
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 2
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001803
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => 0 = 1 - 1
[2]
 => [[1,2]]
 => 0 = 1 - 1
[1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[3]
 => [[1,2,3]]
 => 0 = 1 - 1
[2,1]
 => [[1,3],[2]]
 => 0 = 1 - 1
[1,1,1]
 => [[1],[2],[3]]
 => 2 = 3 - 1
[4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[3,1]
 => [[1,3,4],[2]]
 => 0 = 1 - 1
[2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[2,1,1]
 => [[1,4],[2],[3]]
 => 0 = 1 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3 = 4 - 1
[5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
[4,1]
 => [[1,3,4,5],[2]]
 => 0 = 1 - 1
[3,2]
 => [[1,2,5],[3,4]]
 => 0 = 1 - 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => 0 = 1 - 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => 1 = 2 - 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 0 = 1 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 4 = 5 - 1
[6]
 => [[1,2,3,4,5,6]]
 => 0 = 1 - 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => 0 = 1 - 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => 0 = 1 - 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => 0 = 1 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => 1 = 2 - 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 0 = 1 - 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => 0 = 1 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 2 = 3 - 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 1 = 2 - 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => 0 = 1 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 5 = 6 - 1
[7]
 => [[1,2,3,4,5,6,7]]
 => 0 = 1 - 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => 0 = 1 - 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => 0 = 1 - 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => 0 = 1 - 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => 0 = 1 - 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => 0 = 1 - 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => 0 = 1 - 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 1 = 2 - 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 0 = 1 - 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => 0 = 1 - 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => 0 = 1 - 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 2 = 3 - 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 1 = 2 - 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 6 = 7 - 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St000007
(load all 11 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: 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] => 2
[3]
 => [[1,2,3]]
 => [1,2,3] => 1
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 4
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 1
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 1
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 2
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 5
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 2
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 3
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => 2
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 6
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => 2
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7] => 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => 3
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => 2
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => 7
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000326
(load all 18 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 10 => 1
[2]
 => 100 => 010 => 2
[1,1]
 => 110 => 110 => 1
[3]
 => 1000 => 0010 => 3
[2,1]
 => 1010 => 1100 => 1
[1,1,1]
 => 1110 => 1110 => 1
[4]
 => 10000 => 00010 => 4
[3,1]
 => 10010 => 10100 => 1
[2,2]
 => 1100 => 0110 => 2
[2,1,1]
 => 10110 => 11010 => 1
[1,1,1,1]
 => 11110 => 11110 => 1
[5]
 => 100000 => 000010 => 5
[4,1]
 => 100010 => 100100 => 1
[3,2]
 => 10100 => 01100 => 2
[3,1,1]
 => 100110 => 101010 => 1
[2,2,1]
 => 11010 => 11100 => 1
[2,1,1,1]
 => 101110 => 110110 => 1
[1,1,1,1,1]
 => 111110 => 111110 => 1
[6]
 => 1000000 => 0000010 => 6
[5,1]
 => 1000010 => 1000100 => 1
[4,2]
 => 100100 => 010100 => 2
[4,1,1]
 => 1000110 => 1001010 => 1
[3,3]
 => 11000 => 00110 => 3
[3,2,1]
 => 101010 => 111000 => 1
[3,1,1,1]
 => 1001110 => 1010110 => 1
[2,2,2]
 => 11100 => 01110 => 2
[2,2,1,1]
 => 110110 => 111010 => 1
[2,1,1,1,1]
 => 1011110 => 1101110 => 1
[1,1,1,1,1,1]
 => 1111110 => 1111110 => 1
[7]
 => 10000000 => 00000010 => 7
[6,1]
 => 10000010 => 10000100 => 1
[5,2]
 => 1000100 => 0100100 => 2
[5,1,1]
 => 10000110 => 10001010 => 1
[4,3]
 => 101000 => 001100 => 3
[4,2,1]
 => 1001010 => 1101000 => 1
[4,1,1,1]
 => 10001110 => 10010110 => 1
[3,3,1]
 => 110010 => 101100 => 1
[3,2,2]
 => 101100 => 011010 => 2
[3,2,1,1]
 => 1010110 => 1110010 => 1
[3,1,1,1,1]
 => 10011110 => 10101110 => 1
[2,2,2,1]
 => 111010 => 111100 => 1
[2,2,1,1,1]
 => 1101110 => 1110110 => 1
[2,1,1,1,1,1]
 => 10111110 => 11011110 => 1
[1,1,1,1,1,1,1]
 => 11111110 => 11111110 => 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: St000382
(load all 13 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [1] => 1
[2]
 => [[1,2]]
 => [2] => 2
[1,1]
 => [[1],[2]]
 => [1,1] => 1
[3]
 => [[1,2,3]]
 => [3] => 3
[2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[4]
 => [[1,2,3,4]]
 => [4] => 4
[3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[5]
 => [[1,2,3,4,5]]
 => [5] => 5
[4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
[3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 2
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
[6]
 => [[1,2,3,4,5,6]]
 => [6] => 6
[5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => 2
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 3
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => 2
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => 7
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [1,6] => 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [2,5] => 2
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [1,1,5] => 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [3,4] => 3
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [1,2,4] => 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [1,1,1,4] => 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [1,3,3] => 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [2,2,3] => 2
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [1,1,2,3] => 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [1,1,1,1,3] => 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [1,2,2,2] => 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [1,1,1,2,2] => 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,2] => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 22 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [1] => 1
[2]
 => [[1,2]]
 => [2] => 2
[1,1]
 => [[1],[2]]
 => [1,1] => 1
[3]
 => [[1,2,3]]
 => [3] => 3
[2,1]
 => [[1,2],[3]]
 => [2,1] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[4]
 => [[1,2,3,4]]
 => [4] => 4
[3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[5]
 => [[1,2,3,4,5]]
 => [5] => 5
[4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => 1
[3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => 2
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 1
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 1
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
[6]
 => [[1,2,3,4,5,6]]
 => [6] => 6
[5,1]
 => [[1,2,3,4,5],[6]]
 => [5,1] => 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [4,2] => 2
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [4,1,1] => 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 3
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [3,2,1] => 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [3,1,1,1] => 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => 2
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => 1
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => 7
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [6,1] => 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => 2
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [5,1,1] => 1
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [4,3] => 3
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [4,2,1] => 1
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [4,1,1,1] => 1
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [3,3,1] => 1
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [3,2,2] => 2
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [3,2,1,1] => 1
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [3,1,1,1,1] => 1
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [2,2,2,1] => 1
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [2,2,1,1,1] => 1
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000657
(load all 10 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [1] => 1
[2]
 => [[1,2]]
 => [2] => 2
[1,1]
 => [[1],[2]]
 => [1,1] => 1
[3]
 => [[1,2,3]]
 => [3] => 3
[2,1]
 => [[1,2],[3]]
 => [2,1] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[4]
 => [[1,2,3,4]]
 => [4] => 4
[3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[5]
 => [[1,2,3,4,5]]
 => [5] => 5
[4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => 1
[3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => 2
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 1
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 1
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
[6]
 => [[1,2,3,4,5,6]]
 => [6] => 6
[5,1]
 => [[1,2,3,4,5],[6]]
 => [5,1] => 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [4,2] => 2
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [4,1,1] => 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 3
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [3,2,1] => 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [3,1,1,1] => 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => 2
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => 1
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => 7
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [6,1] => 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => 2
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [5,1,1] => 1
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [4,3] => 3
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [4,2,1] => 1
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [4,1,1,1] => 1
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [3,3,1] => 1
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [3,2,2] => 2
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [3,2,1,1] => 1
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [3,1,1,1,1] => 1
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [2,2,2,1] => 1
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [2,2,1,1,1] => 1
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => 1
Description
The smallest part of an integer composition.
Matching statistic: St000745
(load all 2 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ 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]]
 => 2
[1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
[3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
[4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
[2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
[5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5
[4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 1
[3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 1
[6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
[5,1]
 => [[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 2
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [[1,2,3],[4],[5],[6]]
 => 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 3
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [[1,2,4],[3,5],[6]]
 => 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [[1,2,3,4],[5],[6]]
 => 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 2
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 2
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [[1,2,3],[4],[5],[6],[7]]
 => 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7]]
 => 3
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [[1,2,4],[3,5],[6],[7]]
 => 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [[1,2,3,4],[5],[6],[7]]
 => 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [[1,2,5],[3,6],[4,7]]
 => 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => 2
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6],[7]]
 => 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6],[7]]
 => 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [[1,2,4,6],[3,5,7]]
 => 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [[1,2,3,4,6],[5,7]]
 => 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6],[7]]
 => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[1,2,3,4,5,6,7]]
 => 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000765
(load all 2 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000765: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [1] => 1
[2]
 => [[1,2]]
 => [2] => 1
[1,1]
 => [[1],[2]]
 => [1,1] => 2
[3]
 => [[1,2,3]]
 => [3] => 1
[2,1]
 => [[1,2],[3]]
 => [2,1] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 3
[4]
 => [[1,2,3,4]]
 => [4] => 1
[3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 4
[5]
 => [[1,2,3,4,5]]
 => [5] => 1
[4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => 1
[3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 1
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 5
[6]
 => [[1,2,3,4,5,6]]
 => [6] => 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [5,1] => 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [4,2] => 1
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [4,1,1] => 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 2
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [3,2,1] => 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [3,1,1,1] => 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => 3
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => 2
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 6
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => 1
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [6,1] => 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => 1
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [5,1,1] => 1
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [4,3] => 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [4,2,1] => 1
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [4,1,1,1] => 1
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [3,3,1] => 2
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [3,2,2] => 1
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [3,2,1,1] => 1
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [3,1,1,1,1] => 1
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [2,2,2,1] => 3
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [2,2,1,1,1] => 2
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => 7
Description
The number of weak records in an integer composition. A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$.
The following 340 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000883The number of longest increasing subsequences of a permutation. St001733The number of weak left to right maxima of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000273The domination number of a graph. St000363The number of minimal vertex covers of a graph. St000505The biggest entry in the block containing the 1. St000544The cop number of a graph. St000617The number of global maxima of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000700The protection number of an ordered tree. St000759The smallest missing part in an integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000823The number of unsplittable factors of the set partition. St000911The number of maximal antichains of maximal size in a poset. St000916The packing number of a graph. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001829The common independence number of a graph. St000053The number of valleys of the Dyck path. St000234The number of global ascents of a permutation. St000439The position of the first down step of a Dyck path. St000546The number of global descents of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000969We 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 $c_{n-1}$. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001075The minimal size of a block of a set partition. St000504The cardinality of the first block of a set partition. St000729The minimal arc length of a set partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000974The length of the trunk of an ordered tree. St000989The number of final rises of a permutation. 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. St000054The first entry of the permutation. St000542The number of left-to-right-minima of a permutation. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000918The 2-limited packing number of a graph. St000908The length of the shortest maximal antichain in a poset. St000990The first ascent 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. St000501The size of the first part in the decomposition of a permutation. St000654The first descent of a permutation. St000909The number of maximal chains of maximal size in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000287The number of connected components of a graph. St000740The last entry of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000221The number of strong fixed points of a permutation. St000315The number of isolated vertices of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000756The sum of the positions of the left to right maxima of a permutation. St000667The greatest common divisor of the parts of the partition. St000026The position of the first return of a Dyck path. St000084The number of subtrees. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension 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 $eA$. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St000133The "bounce" of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000120The number of left tunnels of a 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. St000031The number of cycles in the cycle decomposition of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000906The length of the shortest maximal chain in a poset. St001884The number of borders of a binary word. St000633The size of the automorphism group of a poset. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000910The number of maximal chains of minimal length in a poset. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000873The aix statistic of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000210Minimum over maximum difference of elements in cycles. St000338The number of pixed points of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000015The number of peaks of a Dyck path. St000260The radius of a connected graph. St000487The length of the shortest cycle of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001530The depth of a Dyck path. St000331The number of upper interactions of a Dyck path. 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. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000258The burning number of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000781The number of proper colouring schemes of a Ferrers diagram. St001340The cardinality of a minimal non-edge isolating set of a graph. St001432The order dimension of the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St000003The number of standard Young tableaux of the partition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000075The orbit size of a standard tableau under promotion. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000378The diagonal inversion number of an integer partition. St000390The number of runs of ones in a binary word. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000543The size of the conjugacy class of a binary word. St000618The number of self-evacuating tableaux of given shape. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000734The last entry in the first row of a standard tableau. St000783The side length of the largest staircase partition fitting into a partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000983The length of the longest alternating subword. St001006Number of simple modules with projective dimension equal to 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. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001161The major index north count of a Dyck path. 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. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. 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. St001313The number of Dyck paths above the lattice path given by a binary word. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001732The number of peaks visible from the left. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001838The number of nonempty primitive factors of a binary word. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000045The number of linear extensions of a binary tree. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001527The cyclic permutation representation number of an integer partition. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000770The major index of an integer partition when read from bottom to top. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001568The smallest positive integer that does not appear twice in the partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three 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. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000456The monochromatic index of a connected graph. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000406The number of occurrences of the pattern 3241 in a permutation. St000650The number of 3-rises of a permutation. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000454The largest eigenvalue of a graph if it is integral. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000782The indicator function of whether a given perfect matching is an L & P matching. St000929The constant term of the character polynomial of an integer partition. St000942The number of critical left to right maxima of the parking functions. St001330The hat guessing number of a graph. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St001407The number of minimal entries in a semistandard tableau. St001408The number of maximal entries in a semistandard tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001645The pebbling number of a connected graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St000352The Elizalde-Pak rank of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000939The number of characters of the symmetric group whose value on the partition is positive. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000091The descent variation of a composition. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001151The number of blocks with odd minimum. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000089The absolute variation of a composition. St000365The number of double ascents of a permutation. St000562The number of internal points of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000709The number of occurrences of 14-2-3 or 14-3-2. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000839The largest opener of a set partition. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000230Sum of the minimal elements of the blocks of a set partition. St001375The pancake length of a permutation. St001516The number of cyclic bonds of a permutation.