searching the database
Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000086
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 2
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 8
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 12
Description
The number of subgraphs.
Given a graph $G$, this is the number of graphs $H$ such that $H \hookrightarrow G$.
Matching statistic: St000207
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [2]
=> [1,1]
=> 1 = 2 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 7 = 8 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 11 = 12 - 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St001262
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001262: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001262: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> [1]
=> 1 = 2 - 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 7 = 8 - 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 11 = 12 - 1
Description
The dimension of the maximal parabolic seaweed algebra corresponding to the partition.
Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be two compositions of $n$. The corresponding seaweed algebra is the associative subalgebra of the algebra of $n\times n$ matrices which preserves the flags
$$
\{0\} \subset V_1 \subset \cdots \subset V_{m-1} \subset V_m =V
$$
and
$$
V=W_0\supset W_1\supset \cdots \supset W_t=\{0\},
$$
where $V_i=\text{span}\{e_1,\dots, e_{a_1+\cdots +a_i}\}$ and $W_j=\text{span}\{e_{b_1+\cdots +b_j+1},\dots, e_n\}$.
Thus, its dimension is
$$
\frac{1}{2}\left(\sum a_i^2 + \sum b_i^2\right).
$$
It is maximal parabolic if $b_1=n$.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!