Your data matches 43 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
St000529: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 1
00 => 1
01 => 2
10 => 2
11 => 1
000 => 1
001 => 3
010 => 5
011 => 3
100 => 3
101 => 5
110 => 3
111 => 1
Description
The number of permutations whose descent word is the given binary word. This is the sizes of the preimages of the map [[Mp00109]].
Mp00178: Binary words to compositionInteger compositions
St000277: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => 1
1 => [1,1] => 1
00 => [3] => 1
01 => [2,1] => 2
10 => [1,2] => 2
11 => [1,1,1] => 1
000 => [4] => 1
001 => [3,1] => 3
010 => [2,2] => 5
011 => [2,1,1] => 3
100 => [1,3] => 3
101 => [1,2,1] => 5
110 => [1,1,2] => 3
111 => [1,1,1,1] => 1
Description
The number of ribbon shaped standard tableaux. A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape. This is also the size of the preimage of the map 'descent composition' [[Mp00071]] from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed. For a composition $c=c_1,\dots,c_k$ of $n$, the number of ribbon shaped standard tableaux equals $$ \sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell}, $$ where the sum is over all coarsenings of $c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]
Mp00097: Binary words delta morphismInteger compositions
St001102: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [1] => 1
1 => [1] => 1
00 => [2] => 1
01 => [1,1] => 2
10 => [1,1] => 2
11 => [2] => 1
000 => [3] => 1
001 => [2,1] => 3
010 => [1,1,1] => 5
011 => [1,2] => 3
100 => [1,2] => 3
101 => [1,1,1] => 5
110 => [2,1] => 3
111 => [3] => 1
Description
The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. The total number of words with letter multiplicities given by an integer partition is [[St000048]]. For example, there are twelve words with letters $0,0,1,2$ corresponding to the partition $[2,1,1]$. Two of these contain the pattern $132$: $0,0,2,1$ and $0,2,1,0$. Note that this statistic is not constant on compositions having the same parts. The number of words of length $n$ with letters in an alphabet of size $k$ avoiding the consecutive pattern $132$ is determined in [1].
Mp00097: Binary words delta morphismInteger compositions
St001312: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [1] => 1
1 => [1] => 1
00 => [2] => 1
01 => [1,1] => 2
10 => [1,1] => 2
11 => [2] => 1
000 => [3] => 1
001 => [2,1] => 3
010 => [1,1,1] => 5
011 => [1,2] => 3
100 => [1,2] => 3
101 => [1,1,1] => 5
110 => [2,1] => 3
111 => [3] => 1
Description
Number of parabolic noncrossing partitions indexed by the composition. Also the number of elements in the $\nu$-Tamari lattice with $\nu = \nu_\alpha = 1^{\alpha_1} 0^{\alpha_1} \cdots 1^{\alpha_k} 0^{\alpha_k}$, the bounce path indexed by the composition $\alpha$. These elements are Dyck paths weakly above the bounce path $\nu_\alpha$.
Matching statistic: St001103
Mp00097: Binary words delta morphismInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St001103: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [1] => [1]
=> 1
1 => [1] => [1]
=> 1
00 => [2] => [2]
=> 1
01 => [1,1] => [1,1]
=> 2
10 => [1,1] => [1,1]
=> 2
11 => [2] => [2]
=> 1
000 => [3] => [3]
=> 1
001 => [2,1] => [2,1]
=> 3
010 => [1,1,1] => [1,1,1]
=> 5
011 => [1,2] => [2,1]
=> 3
100 => [1,2] => [2,1]
=> 3
101 => [1,1,1] => [1,1,1]
=> 5
110 => [2,1] => [2,1]
=> 3
111 => [3] => [3]
=> 1
Description
The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. The total number of words with letter multiplicities given by an integer partition is [[St000048]]. For example, there are twelve words with letters $0,0,1,2$ corresponding to the partition $[2,1,1]$. Two of these contain an increasing factor of length three: $0012$ and $0120$. Note that prescribing the multiplicities for different letters yields the same number. For example, there are also two words with letters $0,1,1,2$ containing an increasing factor of length three: $1012$ and $0121$. The number of words of length $n$ with letters in an alphabet of size $k$ avoiding the consecutive pattern $123$ is determined in [1].
Matching statistic: St001595
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
St001595: Skew partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> 1
1 => [1,1] => [[1,1],[]]
=> 1
00 => [3] => [[3],[]]
=> 1
01 => [2,1] => [[2,2],[1]]
=> 2
10 => [1,2] => [[2,1],[]]
=> 2
11 => [1,1,1] => [[1,1,1],[]]
=> 1
000 => [4] => [[4],[]]
=> 1
001 => [3,1] => [[3,3],[2]]
=> 3
010 => [2,2] => [[3,2],[1]]
=> 5
011 => [2,1,1] => [[2,2,2],[1,1]]
=> 3
100 => [1,3] => [[3,1],[]]
=> 3
101 => [1,2,1] => [[2,2,1],[1]]
=> 5
110 => [1,1,2] => [[2,1,1],[]]
=> 3
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> 1
Description
The number of standard Young tableaux of the skew partition.
Matching statistic: St000001
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000001: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,3,1] => 1
1 => [1,1] => [1,0,1,0]
=> [3,1,2] => 1
00 => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 1
01 => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2
10 => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 2
11 => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 5
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 3
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 5
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
Description
The number of reduced words for a permutation. This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
Matching statistic: St000100
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00185: Skew partitions cell posetPosets
St000100: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> ([(0,1)],2)
=> 1
1 => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> 1
00 => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
01 => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2
10 => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
11 => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
000 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
001 => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
010 => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 5
011 => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
100 => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
101 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 5
110 => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
Description
The number of linear extensions of a poset.
Matching statistic: St000108
Mp00097: Binary words delta morphismInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> []
=> 1
1 => [1] => [1,0]
=> []
=> 1
00 => [2] => [1,1,0,0]
=> []
=> 1
01 => [1,1] => [1,0,1,0]
=> [1]
=> 2
10 => [1,1] => [1,0,1,0]
=> [1]
=> 2
11 => [2] => [1,1,0,0]
=> []
=> 1
000 => [3] => [1,1,1,0,0,0]
=> []
=> 1
001 => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 3
010 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 5
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 3
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 3
101 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 5
110 => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 3
111 => [3] => [1,1,1,0,0,0]
=> []
=> 1
Description
The number of partitions contained in the given partition.
Matching statistic: St000255
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000255: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,3,1] => 1
1 => [1,1] => [1,0,1,0]
=> [3,1,2] => 1
00 => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 1
01 => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2
10 => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 2
11 => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 5
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 3
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 5
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
Description
The number of reduced Kogan faces with the permutation as type. This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000345The number of refinements of a partition. St000420The number of Dyck paths that are weakly above a Dyck path. St000530The number of permutations with the same descent word as the given permutation. St000532The total number of rook placements on a Ferrers board. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001313The number of Dyck paths above the lattice path given by a binary word. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St001684The reduced word complexity of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001389The number of partitions of the same length below the given integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St000762The sum of the positions of the weak records of an integer composition. St001624The breadth of a lattice. St001644The dimension of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001877Number of indecomposable injective modules with projective dimension 2. St000456The monochromatic index of a connected graph. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000741The Colin de Verdière graph invariant. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001943The sum of the squares of the hook lengths of an integer partition. St000264The girth of a graph, which is not a tree. St000806The semiperimeter of the associated bargraph. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral.