Your data matches 412 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000160
(load all 3 compositions to match this statistic)
Mapping
St000160: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct 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
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)
Mapping
Mp00095: Integer partitions ā€”to binary wordāŸ¶ Binary words
Mp00280: Binary words ā€”path rowmotionāŸ¶ Binary words
St000326: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct 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
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: St000876
(load all 5 compositions to match this statistic)
Mapping
Mp00317: Integer partitions ā€”odd partsāŸ¶ Binary words
Mp00261: Binary words ā€”Burrows-WheelerāŸ¶ Binary words
St000876: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 => 1
[2]
 => 0 => 0 => 1
[1,1]
 => 11 => 11 => 2
[3]
 => 1 => 1 => 1
[2,1]
 => 01 => 10 => 1
[1,1,1]
 => 111 => 111 => 3
[4]
 => 0 => 0 => 1
[3,1]
 => 11 => 11 => 2
[2,2]
 => 00 => 00 => 2
[2,1,1]
 => 011 => 110 => 1
[1,1,1,1]
 => 1111 => 1111 => 4
[5]
 => 1 => 1 => 1
[4,1]
 => 01 => 10 => 1
[3,2]
 => 10 => 10 => 1
[3,1,1]
 => 111 => 111 => 3
[2,2,1]
 => 001 => 100 => 1
[2,1,1,1]
 => 0111 => 1110 => 2
[1,1,1,1,1]
 => 11111 => 11111 => 5
[6]
 => 0 => 0 => 1
[5,1]
 => 11 => 11 => 2
[4,2]
 => 00 => 00 => 2
[4,1,1]
 => 011 => 110 => 1
[3,3]
 => 11 => 11 => 2
[3,2,1]
 => 101 => 110 => 1
[3,1,1,1]
 => 1111 => 1111 => 4
[2,2,2]
 => 000 => 000 => 3
[2,2,1,1]
 => 0011 => 1010 => 1
[2,1,1,1,1]
 => 01111 => 11110 => 3
[1,1,1,1,1,1]
 => 111111 => 111111 => 6
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: St001088
(load all 7 compositions to match this statistic)
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00132: Dyck paths ā€”switch returns and last double riseāŸ¶ Dyck paths
St001088: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 6
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 4
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 1
Description
Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.
Matching statistic: St000877
Mapping
Mp00095: Integer partitions ā€”to binary wordāŸ¶ Binary words
Mp00280: Binary words ā€”path rowmotionāŸ¶ Binary words
St000877: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 11 => 0 = 1 - 1
[2]
 => 100 => 011 => 1 = 2 - 1
[1,1]
 => 110 => 111 => 0 = 1 - 1
[3]
 => 1000 => 0011 => 2 = 3 - 1
[2,1]
 => 1010 => 1101 => 0 = 1 - 1
[1,1,1]
 => 1110 => 1111 => 0 = 1 - 1
[4]
 => 10000 => 00011 => 3 = 4 - 1
[3,1]
 => 10010 => 01101 => 1 = 2 - 1
[2,2]
 => 1100 => 0111 => 1 = 2 - 1
[2,1,1]
 => 10110 => 11011 => 0 = 1 - 1
[1,1,1,1]
 => 11110 => 11111 => 0 = 1 - 1
[5]
 => 100000 => 000011 => 4 = 5 - 1
[4,1]
 => 100010 => 001101 => 2 = 3 - 1
[3,2]
 => 10100 => 11001 => 0 = 1 - 1
[3,1,1]
 => 100110 => 011011 => 1 = 2 - 1
[2,2,1]
 => 11010 => 11101 => 0 = 1 - 1
[2,1,1,1]
 => 101110 => 110111 => 0 = 1 - 1
[1,1,1,1,1]
 => 111110 => 111111 => 0 = 1 - 1
[6]
 => 1000000 => 0000011 => 5 = 6 - 1
[5,1]
 => 1000010 => 0001101 => 3 = 4 - 1
[4,2]
 => 100100 => 011001 => 1 = 2 - 1
[4,1,1]
 => 1000110 => 0011011 => 2 = 3 - 1
[3,3]
 => 11000 => 00111 => 2 = 3 - 1
[3,2,1]
 => 101010 => 110101 => 0 = 1 - 1
[3,1,1,1]
 => 1001110 => 0110111 => 1 = 2 - 1
[2,2,2]
 => 11100 => 01111 => 1 = 2 - 1
[2,2,1,1]
 => 110110 => 111011 => 0 = 1 - 1
[2,1,1,1,1]
 => 1011110 => 1101111 => 0 = 1 - 1
[1,1,1,1,1,1]
 => 1111110 => 1111111 => 0 = 1 - 1
Description
The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St000011
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00327: Dyck paths ā€”inverse Kreweras complementāŸ¶ Dyck paths
Mp00124: Dyck paths ā€”Adin-Bagno-Roichman transformationāŸ¶ Dyck paths
St000011: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct 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
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: St000153
Mapping
Mp00045: Integer partitions ā€”reading tableauāŸ¶ Standard tableaux
Mp00284: Standard tableaux ā€”rowsāŸ¶ Set partitions
Mp00080: Set partitions ā€”to permutationāŸ¶ Permutations
St000153: Permutations āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => {{1}}
 => [1] => 1
[2]
 => [[1,2]]
 => {{1,2}}
 => [2,1] => 1
[1,1]
 => [[1],[2]]
 => {{1},{2}}
 => [1,2] => 2
[3]
 => [[1,2,3]]
 => {{1,2,3}}
 => [2,3,1] => 1
[2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => [3,2,1] => 1
[1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => [1,2,3] => 3
[4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => [2,3,4,1] => 1
[3,1]
 => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => [3,2,4,1] => 1
[2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => [2,1,4,3] => 2
[2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 4
[5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 1
[4,1]
 => [[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => [3,2,4,5,1] => 1
[3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => [2,5,4,3,1] => 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => [4,2,3,5,1] => 2
[2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => 3
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => 5
[6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => {{1,3,4,5,6},{2}}
 => [3,2,4,5,6,1] => 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => [2,5,4,3,6,1] => 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => {{1,4,5,6},{2},{3}}
 => [4,2,3,5,6,1] => 2
[3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => 2
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => {{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => {{1,5,6},{2},{3},{4}}
 => [5,2,3,4,6,1] => 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => {{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => 3
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => {{1,4},{2,6},{3},{5}}
 => [4,6,3,1,5,2] => 2
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => {{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => 4
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => 6
Description
The number of adjacent cycles of a permutation. This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000374
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
Mp00024: Dyck paths ā€”to 321-avoiding permutationāŸ¶ Permutations
Mp00241: Permutations ā€”invert Laguerre heapāŸ¶ Permutations
St000374: Permutations āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2,1] => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => [2,1,3] => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => 3
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,3,1,4] => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,4,1,3] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [3,1,2,4] => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 4
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [2,3,4,1,5] => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [4,2,3,1] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,4,1,2] => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 5
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => [2,3,4,5,1,6] => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [2,5,3,4,1] => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,3,1,4,5] => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,3,5,1,4] => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => 2
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,5,1,3,4] => 3
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,5,1,2,3] => 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,1,6] => [5,1,2,3,4,6] => 4
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => 6
Description
The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].
Matching statistic: St000382
(load all 8 compositions to match this statistic)
Mapping
Mp00045: Integer partitions ā€”reading tableauāŸ¶ Standard tableaux
Mp00207: Standard tableaux ā€”horizontal strip sizesāŸ¶ Integer compositions
Mp00133: Integer compositions ā€”delta morphismāŸ¶ Integer compositions
St000382: Integer compositions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct 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
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 7 compositions to match this statistic)
Mapping
Mp00042: Integer partitions ā€”initial tableauāŸ¶ Standard tableaux
Mp00207: Standard tableaux ā€”horizontal strip sizesāŸ¶ Integer compositions
Mp00133: Integer compositions ā€”delta morphismāŸ¶ Integer compositions
St000383: Integer compositions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct 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
Description
The last part of an integer composition.
The following 402 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000678The number of up steps after the last double rise of a Dyck path. St000703The number of deficiencies of a permutation. St000899The maximal number of repetitions of an integer composition. St000907The number of maximal antichains of minimal length in a poset. St000932The number of occurrences of the pattern UDU in a Dyck path. St000996The number of exclusive left-to-right maxima of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nāˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001733The number of weak left to right maxima of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000058The order of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000297The number of leading ones in a binary word. St000356The number of occurrences of the pattern 13-2. St000366The number of double descents of a permutation. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000648The number of 2-excedences of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000731The number of double exceedences of a permutation. St000982The length of the longest constant subword. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: 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 $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St000306The bounce count of a Dyck path. St000654The first descent of a permutation. St000717The number of ordinal summands of a poset. St000910The number of maximal chains of minimal length in a poset. St000989The number of final rises of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000444The length of the maximal rise of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000485The length of the longest cycle of a permutation. St000740The last entry of a permutation. St000873The aix statistic of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001061The number of indices that are both descents and recoils of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000439The position of the first down step of a Dyck path. 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. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. 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. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000335The difference of lower and upper interactions. 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. St000864The number of circled entries of the shifted recording tableau of a permutation. St000990The first ascent 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. 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 $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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 $eA$. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. 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. St000066The column of the unique '1' in the first row of the alternating sign matrix. 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. St000317The cycle descent number of a permutation. St000331The number of upper interactions of a Dyck path. St000358The number of occurrences of the pattern 31-2. St000461The rix statistic of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000732The number of double deficiencies of a 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. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. 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. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001552The number of inversions between excedances and fixed points of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves 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. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. 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. St001811The Castelnuovo-Mumford regularity of a permutation. 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000744The length of the path to the largest entry in a standard Young tableau. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000060The greater neighbor of the maximum. St000619The number of cyclic descents of a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000933The number of multipartitions of sizes given by an integer partition. St001052The length of the exterior of a permutation. St001128The exponens consonantiae of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St001884The number of borders of a binary word. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001438The number of missing boxes of a skew partition. St001868The number of alignments of type NE of a signed permutation. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000456The monochromatic index of a connected graph. St000667The greatest common divisor of the parts of the partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St000765The number of weak records in an integer composition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001432The order dimension of the partition. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. 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. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. 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. St000048The multinomial of the parts of a partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000228The size of a partition. St000288The number of ones in a binary word. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000443The number of long tunnels of a Dyck path. St000459The hook length of the base cell of a partition. St000519The largest length of a factor maximising the subword complexity. 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. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000733The row containing the largest entry of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. 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. St000867The sum of the hook lengths in the first row of an integer partition. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. 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. St001400The total number of Littlewood-Richardson tableaux of given shape. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001471The magnitude of a Dyck path. St001485The modular major index of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001614The cyclic permutation representation number of a skew partition. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001959The product of the heights of the peaks of a Dyck path. 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 $K[x]/(x^n)$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. 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. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000454The largest eigenvalue of a graph if it is integral. St000993The multiplicity of the largest part of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. 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. 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 $eA$. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001330The hat guessing number of a graph. St001118The acyclic chromatic index of a graph. St000284The Plancherel distribution on integer partitions. 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. St000770The major index of an integer partition when read from bottom to top. St000782The indicator function of whether a given perfect matching is an L & P matching. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001568The smallest positive integer that does not appear twice in the partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St000515The number of invariant set partitions when acting with a permutation of given cycle type. 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. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001808The box weight or horizontal decoration of a Dyck path. St000259The diameter of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. 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. St001780The order of promotion on the set of standard tableaux of given shape. 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. 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. St001556The number of inversions of the third entry of a permutation. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001645The pebbling number of a connected graph. St001487The number of inner corners of a skew partition. 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. St000260The radius of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. 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. St001060The distinguishing index of a graph. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001151The number of blocks with odd minimum. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000632The jump number of the poset. St000736The last entry in the first row of a semistandard tableau. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000307The number of rowmotion orbits of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001520The number of strict 3-descents.