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Your data matches 32 different statistics following compositions of up to 3 maps.
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Matching statistic: St001814
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001814: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001814: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 3 = 2 + 1
([],2)
=> [2,2]
=> 3 = 2 + 1
([(0,1)],2)
=> [3]
=> 4 = 3 + 1
([],3)
=> [2,2,2,2]
=> 3 = 2 + 1
([(1,2)],3)
=> [6]
=> 7 = 6 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> 6 = 5 + 1
([(0,2),(2,1)],3)
=> [4]
=> 5 = 4 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> 6 = 5 + 1
Description
The number of partitions interlacing the given partition.
Matching statistic: St001918
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 1 = 2 - 1
([],2)
=> [2,2]
=> 1 = 2 - 1
([(0,1)],2)
=> [3]
=> 2 = 3 - 1
([],3)
=> [2,2,2,2]
=> 1 = 2 - 1
([(1,2)],3)
=> [6]
=> 5 = 6 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> 4 = 5 - 1
([(0,2),(2,1)],3)
=> [4]
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> 4 = 5 - 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
Matching statistic: St000548
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1]
=> 2
([],2)
=> [2,2]
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 3
([],3)
=> [2,2,2,2]
=> [4,4]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 5
Description
The number of different non-empty partial sums of an integer partition.
Matching statistic: St000734
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [[1,2]]
=> 2
([],2)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,1)],2)
=> [3]
=> [[1,2,3]]
=> 3
([],3)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 2
([(1,2)],3)
=> [6]
=> [[1,2,3,4,5,6]]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 5
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000947
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Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5
Description
The major index east count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
Matching statistic: St001161
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5
Description
The major index north count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index north count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$.
Matching statistic: St000012
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,2)],3)
=> [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]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 5
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000738
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
([],2)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
([(0,1)],2)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
([],3)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
([(1,2)],3)
=> [6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 5
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St001809
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
Description
The index of the step at the first peak of maximal height in a Dyck path.
Matching statistic: St000005
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 88%●distinct values known / distinct values provided: 80%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 88%●distinct values known / distinct values provided: 80%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 5
Description
The bounce statistic of a Dyck path.
The '''bounce path''' $D'$ of a Dyck path $D$ is the Dyck path obtained from $D$ by starting at the end point $(2n,0)$, traveling north-west until hitting $D$, then bouncing back south-west to the $x$-axis, and repeating this procedure until finally reaching the point $(0,0)$.
The points where $D'$ touches the $x$-axis are called '''bounce points''', and a bounce path is uniquely determined by its bounce points.
This statistic is given by the sum of all $i$ for which the bounce path $D'$ of $D$ touches the $x$-axis at $(2i,0)$.
In particular, the bounce statistics of $D$ and $D'$ coincide.
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St001645The pebbling number of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001545The second Elser number of a connected graph. St000180The number of chains of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St001909The number of interval-closed sets of a poset. St000656The number of cuts of a poset. St000707The product of the factorials of the parts. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. 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. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph.
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