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Your data matches 299 different statistics following compositions of up to 3 maps.
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Matching statistic: St000160
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000160: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 3
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 4
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 5
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 2
[3,3]
=> 2
[3,2,1]
=> 1
[3,1,1,1]
=> 3
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 6
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 2
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 3
[3,3,1]
=> 1
[3,2,2]
=> 2
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 4
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 7
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000326
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00095: Integer partitions āto binary wordā¶ Binary words
Mp00280: Binary words āpath rowmotionā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00280: Binary words āpath rowmotionā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 11 => 1
[2]
=> 100 => 011 => 2
[1,1]
=> 110 => 111 => 1
[3]
=> 1000 => 0011 => 3
[2,1]
=> 1010 => 1101 => 1
[1,1,1]
=> 1110 => 1111 => 1
[4]
=> 10000 => 00011 => 4
[3,1]
=> 10010 => 01101 => 2
[2,2]
=> 1100 => 0111 => 2
[2,1,1]
=> 10110 => 11011 => 1
[1,1,1,1]
=> 11110 => 11111 => 1
[5]
=> 100000 => 000011 => 5
[4,1]
=> 100010 => 001101 => 3
[3,2]
=> 10100 => 11001 => 1
[3,1,1]
=> 100110 => 011011 => 2
[2,2,1]
=> 11010 => 11101 => 1
[2,1,1,1]
=> 101110 => 110111 => 1
[1,1,1,1,1]
=> 111110 => 111111 => 1
[6]
=> 1000000 => 0000011 => 6
[5,1]
=> 1000010 => 0001101 => 4
[4,2]
=> 100100 => 011001 => 2
[4,1,1]
=> 1000110 => 0011011 => 3
[3,3]
=> 11000 => 00111 => 3
[3,2,1]
=> 101010 => 110101 => 1
[3,1,1,1]
=> 1001110 => 0110111 => 2
[2,2,2]
=> 11100 => 01111 => 2
[2,2,1,1]
=> 110110 => 111011 => 1
[2,1,1,1,1]
=> 1011110 => 1101111 => 1
[1,1,1,1,1,1]
=> 1111110 => 1111111 => 1
[7]
=> 10000000 => 00000011 => 7
[6,1]
=> 10000010 => 00001101 => 5
[5,2]
=> 1000100 => 0011001 => 3
[5,1,1]
=> 10000110 => 00011011 => 4
[4,3]
=> 101000 => 110001 => 1
[4,2,1]
=> 1001010 => 0110101 => 2
[4,1,1,1]
=> 10001110 => 00110111 => 3
[3,3,1]
=> 110010 => 011101 => 2
[3,2,2]
=> 101100 => 110011 => 1
[3,2,1,1]
=> 1010110 => 1101011 => 1
[3,1,1,1,1]
=> 10011110 => 01101111 => 2
[2,2,2,1]
=> 111010 => 111101 => 1
[2,2,1,1,1]
=> 1101110 => 1110111 => 1
[2,1,1,1,1,1]
=> 10111110 => 11011111 => 1
[1,1,1,1,1,1,1]
=> 11111110 => 11111111 => 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: St000011
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00327: Dyck paths āinverse Kreweras complementā¶ Dyck paths
Mp00124: Dyck paths āAdin-Bagno-Roichman transformationā¶ Dyck paths
St000011: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00327: Dyck paths āinverse Kreweras complementā¶ Dyck paths
Mp00124: Dyck paths āAdin-Bagno-Roichman transformationā¶ Dyck paths
St000011: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 4
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 5
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 7
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: St000382
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00207: Standard tableaux āhorizontal strip sizesā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
St000382: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00207: Standard tableaux āhorizontal strip sizesā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
St000382: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [2] => [1] => 1
[1,1]
=> [[1],[2]]
=> [1,1] => [2] => 2
[3]
=> [[1,2,3]]
=> [3] => [1] => 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => [1,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 3
[4]
=> [[1,2,3,4]]
=> [4] => [1] => 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => [1,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2] => 2
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => [2,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 4
[5]
=> [[1,2,3,4,5]]
=> [5] => [1] => 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => [1,1] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => [1,1] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => [2,1] => 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [1,2] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => [3,1] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => [1,1] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => [1,1] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => [2,1] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [2] => 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => [1,1,1] => 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => [3,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [3] => 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => [2,2] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => [4,1] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [6] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [1] => 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => [1,1] => 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => [1,1] => 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => [2,1] => 2
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => [1,1] => 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => [1,1,1] => 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => [3,1] => 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => [1,2] => 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => [2,1] => 2
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => [2,1,1] => 2
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => [4,1] => 4
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => [1,3] => 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => [3,2] => 3
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => [5,1] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [7] => 7
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00207: Standard tableaux āhorizontal strip sizesā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
St000383: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00207: Standard tableaux āhorizontal strip sizesā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
St000383: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [2] => [1] => 1
[1,1]
=> [[1],[2]]
=> [1,1] => [2] => 2
[3]
=> [[1,2,3]]
=> [3] => [1] => 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => [1,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 3
[4]
=> [[1,2,3,4]]
=> [4] => [1] => 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => [1,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2] => 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 4
[5]
=> [[1,2,3,4,5]]
=> [5] => [1] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,1] => 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => [1,1] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,2] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [2,1] => 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,3] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => [1,1] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,2] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [2] => 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => [1,1,1] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,3] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [3] => 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [2,2] => 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,4] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [6] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [1] => 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => [1,1] => 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => [1,1] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,2] => 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => [1,1] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => [1,1,1] => 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,3] => 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => [2,1] => 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => [1,2] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => [1,1,2] => 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => [1,4] => 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => [3,1] => 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => [2,3] => 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => [1,5] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [7] => 7
Description
The last part of an integer composition.
Matching statistic: St000876
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00317: Integer partitions āodd partsā¶ Binary words
Mp00224: Binary words ārunsortā¶ Binary words
Mp00104: Binary words āreverseā¶ Binary words
St000876: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00224: Binary words ārunsortā¶ Binary words
Mp00104: Binary words āreverseā¶ Binary words
St000876: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 1 => 1
[2]
=> 0 => 0 => 0 => 1
[1,1]
=> 11 => 11 => 11 => 2
[3]
=> 1 => 1 => 1 => 1
[2,1]
=> 01 => 01 => 10 => 1
[1,1,1]
=> 111 => 111 => 111 => 3
[4]
=> 0 => 0 => 0 => 1
[3,1]
=> 11 => 11 => 11 => 2
[2,2]
=> 00 => 00 => 00 => 2
[2,1,1]
=> 011 => 011 => 110 => 1
[1,1,1,1]
=> 1111 => 1111 => 1111 => 4
[5]
=> 1 => 1 => 1 => 1
[4,1]
=> 01 => 01 => 10 => 1
[3,2]
=> 10 => 01 => 10 => 1
[3,1,1]
=> 111 => 111 => 111 => 3
[2,2,1]
=> 001 => 001 => 100 => 1
[2,1,1,1]
=> 0111 => 0111 => 1110 => 2
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[6]
=> 0 => 0 => 0 => 1
[5,1]
=> 11 => 11 => 11 => 2
[4,2]
=> 00 => 00 => 00 => 2
[4,1,1]
=> 011 => 011 => 110 => 1
[3,3]
=> 11 => 11 => 11 => 2
[3,2,1]
=> 101 => 011 => 110 => 1
[3,1,1,1]
=> 1111 => 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 000 => 3
[2,2,1,1]
=> 0011 => 0011 => 1100 => 1
[2,1,1,1,1]
=> 01111 => 01111 => 11110 => 3
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 6
[7]
=> 1 => 1 => 1 => 1
[6,1]
=> 01 => 01 => 10 => 1
[5,2]
=> 10 => 01 => 10 => 1
[5,1,1]
=> 111 => 111 => 111 => 3
[4,3]
=> 01 => 01 => 10 => 1
[4,2,1]
=> 001 => 001 => 100 => 1
[4,1,1,1]
=> 0111 => 0111 => 1110 => 2
[3,3,1]
=> 111 => 111 => 111 => 3
[3,2,2]
=> 100 => 001 => 100 => 1
[3,2,1,1]
=> 1011 => 0111 => 1110 => 2
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[2,2,2,1]
=> 0001 => 0001 => 1000 => 2
[2,2,1,1,1]
=> 00111 => 00111 => 11100 => 1
[2,1,1,1,1,1]
=> 011111 => 011111 => 111110 => 4
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 7
Description
The number of factors in the Catalan decomposition of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St000899
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00317: Integer partitions āodd partsā¶ Binary words
Mp00097: Binary words ādelta morphismā¶ Integer compositions
Mp00039: Integer compositions ācomplementā¶ Integer compositions
St000899: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00097: Binary words ādelta morphismā¶ Integer compositions
Mp00039: Integer compositions ācomplementā¶ Integer compositions
St000899: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1] => [1] => 1
[2]
=> 0 => [1] => [1] => 1
[1,1]
=> 11 => [2] => [1,1] => 2
[3]
=> 1 => [1] => [1] => 1
[2,1]
=> 01 => [1,1] => [2] => 1
[1,1,1]
=> 111 => [3] => [1,1,1] => 3
[4]
=> 0 => [1] => [1] => 1
[3,1]
=> 11 => [2] => [1,1] => 2
[2,2]
=> 00 => [2] => [1,1] => 2
[2,1,1]
=> 011 => [1,2] => [2,1] => 1
[1,1,1,1]
=> 1111 => [4] => [1,1,1,1] => 4
[5]
=> 1 => [1] => [1] => 1
[4,1]
=> 01 => [1,1] => [2] => 1
[3,2]
=> 10 => [1,1] => [2] => 1
[3,1,1]
=> 111 => [3] => [1,1,1] => 3
[2,2,1]
=> 001 => [2,1] => [1,2] => 1
[2,1,1,1]
=> 0111 => [1,3] => [2,1,1] => 2
[1,1,1,1,1]
=> 11111 => [5] => [1,1,1,1,1] => 5
[6]
=> 0 => [1] => [1] => 1
[5,1]
=> 11 => [2] => [1,1] => 2
[4,2]
=> 00 => [2] => [1,1] => 2
[4,1,1]
=> 011 => [1,2] => [2,1] => 1
[3,3]
=> 11 => [2] => [1,1] => 2
[3,2,1]
=> 101 => [1,1,1] => [3] => 1
[3,1,1,1]
=> 1111 => [4] => [1,1,1,1] => 4
[2,2,2]
=> 000 => [3] => [1,1,1] => 3
[2,2,1,1]
=> 0011 => [2,2] => [1,2,1] => 1
[2,1,1,1,1]
=> 01111 => [1,4] => [2,1,1,1] => 3
[1,1,1,1,1,1]
=> 111111 => [6] => [1,1,1,1,1,1] => 6
[7]
=> 1 => [1] => [1] => 1
[6,1]
=> 01 => [1,1] => [2] => 1
[5,2]
=> 10 => [1,1] => [2] => 1
[5,1,1]
=> 111 => [3] => [1,1,1] => 3
[4,3]
=> 01 => [1,1] => [2] => 1
[4,2,1]
=> 001 => [2,1] => [1,2] => 1
[4,1,1,1]
=> 0111 => [1,3] => [2,1,1] => 2
[3,3,1]
=> 111 => [3] => [1,1,1] => 3
[3,2,2]
=> 100 => [1,2] => [2,1] => 1
[3,2,1,1]
=> 1011 => [1,1,2] => [3,1] => 1
[3,1,1,1,1]
=> 11111 => [5] => [1,1,1,1,1] => 5
[2,2,2,1]
=> 0001 => [3,1] => [1,1,2] => 2
[2,2,1,1,1]
=> 00111 => [2,3] => [1,2,1,1] => 2
[2,1,1,1,1,1]
=> 011111 => [1,5] => [2,1,1,1,1] => 4
[1,1,1,1,1,1,1]
=> 1111111 => [7] => [1,1,1,1,1,1,1] => 7
Description
The maximal number of repetitions of an integer composition.
This is the maximal part of the composition obtained by applying the delta morphism.
Matching statistic: St000907
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00227: Dyck paths āDelest-Viennot-inverseā¶ Dyck paths
Mp00242: Dyck paths āHessenberg posetā¶ Posets
St000907: Posets ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00227: Dyck paths āDelest-Viennot-inverseā¶ Dyck paths
Mp00242: Dyck paths āHessenberg posetā¶ Posets
St000907: Posets ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> ([],1)
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([],6)
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([],7)
=> 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5)],6)
=> 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ([(0,6),(6,1),(6,2),(6,3),(6,4),(6,5)],7)
=> 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ([(0,5),(5,6),(6,1),(6,2),(6,3),(6,4)],7)
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000297
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00095: Integer partitions āto binary wordā¶ Binary words
Mp00280: Binary words āpath rowmotionā¶ Binary words
Mp00105: Binary words ācomplementā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00280: Binary words āpath rowmotionā¶ Binary words
Mp00105: Binary words ācomplementā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 11 => 00 => 0 = 1 - 1
[2]
=> 100 => 011 => 100 => 1 = 2 - 1
[1,1]
=> 110 => 111 => 000 => 0 = 1 - 1
[3]
=> 1000 => 0011 => 1100 => 2 = 3 - 1
[2,1]
=> 1010 => 1101 => 0010 => 0 = 1 - 1
[1,1,1]
=> 1110 => 1111 => 0000 => 0 = 1 - 1
[4]
=> 10000 => 00011 => 11100 => 3 = 4 - 1
[3,1]
=> 10010 => 01101 => 10010 => 1 = 2 - 1
[2,2]
=> 1100 => 0111 => 1000 => 1 = 2 - 1
[2,1,1]
=> 10110 => 11011 => 00100 => 0 = 1 - 1
[1,1,1,1]
=> 11110 => 11111 => 00000 => 0 = 1 - 1
[5]
=> 100000 => 000011 => 111100 => 4 = 5 - 1
[4,1]
=> 100010 => 001101 => 110010 => 2 = 3 - 1
[3,2]
=> 10100 => 11001 => 00110 => 0 = 1 - 1
[3,1,1]
=> 100110 => 011011 => 100100 => 1 = 2 - 1
[2,2,1]
=> 11010 => 11101 => 00010 => 0 = 1 - 1
[2,1,1,1]
=> 101110 => 110111 => 001000 => 0 = 1 - 1
[1,1,1,1,1]
=> 111110 => 111111 => 000000 => 0 = 1 - 1
[6]
=> 1000000 => 0000011 => 1111100 => 5 = 6 - 1
[5,1]
=> 1000010 => 0001101 => 1110010 => 3 = 4 - 1
[4,2]
=> 100100 => 011001 => 100110 => 1 = 2 - 1
[4,1,1]
=> 1000110 => 0011011 => 1100100 => 2 = 3 - 1
[3,3]
=> 11000 => 00111 => 11000 => 2 = 3 - 1
[3,2,1]
=> 101010 => 110101 => 001010 => 0 = 1 - 1
[3,1,1,1]
=> 1001110 => 0110111 => 1001000 => 1 = 2 - 1
[2,2,2]
=> 11100 => 01111 => 10000 => 1 = 2 - 1
[2,2,1,1]
=> 110110 => 111011 => 000100 => 0 = 1 - 1
[2,1,1,1,1]
=> 1011110 => 1101111 => 0010000 => 0 = 1 - 1
[1,1,1,1,1,1]
=> 1111110 => 1111111 => 0000000 => 0 = 1 - 1
[7]
=> 10000000 => 00000011 => 11111100 => 6 = 7 - 1
[6,1]
=> 10000010 => 00001101 => 11110010 => 4 = 5 - 1
[5,2]
=> 1000100 => 0011001 => 1100110 => 2 = 3 - 1
[5,1,1]
=> 10000110 => 00011011 => 11100100 => 3 = 4 - 1
[4,3]
=> 101000 => 110001 => 001110 => 0 = 1 - 1
[4,2,1]
=> 1001010 => 0110101 => 1001010 => 1 = 2 - 1
[4,1,1,1]
=> 10001110 => 00110111 => 11001000 => 2 = 3 - 1
[3,3,1]
=> 110010 => 011101 => 100010 => 1 = 2 - 1
[3,2,2]
=> 101100 => 110011 => 001100 => 0 = 1 - 1
[3,2,1,1]
=> 1010110 => 1101011 => 0010100 => 0 = 1 - 1
[3,1,1,1,1]
=> 10011110 => 01101111 => 10010000 => 1 = 2 - 1
[2,2,2,1]
=> 111010 => 111101 => 000010 => 0 = 1 - 1
[2,2,1,1,1]
=> 1101110 => 1110111 => 0001000 => 0 = 1 - 1
[2,1,1,1,1,1]
=> 10111110 => 11011111 => 00100000 => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> 11111110 => 11111111 => 00000000 => 0 = 1 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000648
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
St000648: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
St000648: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 3 = 4 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 4 = 5 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => 5 = 6 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => 3 = 4 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2 = 3 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => 0 = 1 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,2] => 6 = 7 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,2,3] => 4 = 5 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => 2 = 3 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => 3 = 4 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 0 = 1 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2 = 3 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,2,5,6,4] => 1 = 2 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,3] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => 0 = 1 - 1
Description
The number of 2-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
The following 289 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000731The number of double exceedences of a permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000237The number of small exceedances. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000058The order of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000439The position of the first down step of a Dyck path. St000654The first descent of a permutation. St000717The number of ordinal summands of a poset. St000996The number of exclusive left-to-right maxima of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000740The last entry of a permutation. St000873The aix statistic of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000485The length of the longest cycle of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000314The number of left-to-right-maxima of a permutation. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000338The number of pixed points 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000051The size of the left subtree of a binary tree. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000352The Elizalde-Pak rank of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000990The first ascent of a permutation. St001050The number of terminal closers of a set partition. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000039The number of crossings of a permutation. St000054The first entry of the permutation. St000117The number of centered tunnels of a Dyck path. St000203The number of external nodes of a binary tree. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St000331The number of upper interactions of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000734The last entry in the first row of a standard tableau. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000971The smallest closer of a set partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. 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. 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. 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. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001346The number of parking functions that give the same permutation. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001530The depth of a Dyck path. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000756The sum of the positions of the left to right maxima of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001061The number of indices that are both descents and recoils of a permutation. St000153The number of adjacent cycles of a permutation. St000910The number of maximal chains of minimal length in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001052The length of the exterior of a permutation. St000007The number of saliances of the permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000461The rix statistic 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. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001557The number of inversions of the second entry of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000738The first entry in the last row of a standard tableau. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St000667The greatest common divisor of the parts of the partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001527The cyclic permutation representation number of an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000010The length of the partition. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000288The number of ones in a binary word. St000335The difference of lower and upper interactions. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000548The number of different non-empty partial sums of an integer partition. St000627The exponent of a binary word. St000733The row containing the largest entry of a standard tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of 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. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001372The length of a longest cyclic run of ones of a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001523The degree of symmetry of a Dyck path. St001614The cyclic permutation representation number of a skew partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001884The number of borders of a binary word. St001959The product of the heights of the peaks of a Dyck path. St000456The monochromatic index of a connected graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000993The multiplicity of the largest part of an integer partition. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001118The acyclic chromatic index of a graph. St001438The number of missing boxes of a skew partition. St001868The number of alignments of type NE of a signed permutation. 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. 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. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000284The Plancherel distribution on integer partitions. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000744The length of the path to the largest entry in a standard Young tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. 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. St001128The exponens consonantiae of a partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001435The number of missing boxes in the first row. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000765The number of weak records in an integer composition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001570The minimal number of edges to add to make a graph Hamiltonian. St001730The number of times the path corresponding to a binary word crosses the base line. St001948The number of augmented double ascents of a permutation. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000657The smallest part of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. 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. St000516The number of stretching pairs of a permutation. St000650The number of 3-rises of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001847The number of occurrences of the pattern 1432 in a permutation. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000259The diameter of a connected graph. St001330The hat guessing number of a graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000455The second largest eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000260The radius 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. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001060The distinguishing index of a graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001487The number of inner corners of a skew partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000090The variation of a composition. 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. St001722The number of minimal chains with small intervals between a binary word and the top element. 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. St000075The orbit size of a standard tableau under promotion. 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. 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. 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. St000735The last entry on the main diagonal of a standard tableau.
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