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Your data matches 171 different statistics following compositions of up to 3 maps.
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Matching statistic: St001414
(load all 19 compositions to match this statistic)

Mapping

St001414: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 0
=> 0
=> 0
=> 0
=> 0
=> 0
=> 1
=> 0
=> 1
=> 0
=> 0
=> 1
=> 0
=> 1
=> 1
=> 1
=> 0
=> 0
=> 1
=> 1
=> 0
=> 0
=> 0
=> 0
=> 1
=> 1
=> 0
=> 0
=> 1
=> 1

Description

Half the length of the longest odd length palindromic prefix of a binary word. More precisely, this statistic is the largest number

$k$ such that the word has a palindromic prefix of length

$2k+1$ .

Matching statistic: St001263
(load all 2 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
St001263: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> [1] => 0
=> [1] => 0
=> [2] => 0
=> [1,1] => 0
=> [1,1] => 0
=> [2] => 0
=> [3] => 1
=> [2,1] => 0
=> [1,1,1] => 1
=> [1,2] => 0
=> [1,2] => 0
=> [1,1,1] => 1
=> [2,1] => 0
=> [3] => 1
=> [4] => 1
=> [3,1] => 0
=> [2,1,1] => 0
=> [2,2] => 1
=> [1,1,2] => 0
=> [1,1,1,1] => 1
=> [1,2,1] => 1
=> [1,3] => 0
=> [1,3] => 0
=> [1,2,1] => 1
=> [1,1,1,1] => 1
=> [1,1,2] => 0
=> [2,2] => 1
=> [2,1,1] => 0
=> [3,1] => 0
=> [4] => 1

Description

The index of the maximal parabolic seaweed algebra associated with the composition. Let

$a_1,\dots,a_m$ and

$b_1,\dots,b_t$ be a pair of compositions of

$n$ . The meander associated to this pair is obtained as follows: * place

$n$ dots on a horizontal line * subdivide the dots into

$m$ blocks of sizes

$a_1, a_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc above the line * subdivide the dots into

$t$ blocks of sizes

$b_1, b_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc below the line By [1, thm.5.1], the index of the seaweed algebra associated to the pair of compositions is

$\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{a_1|a_2|...|a_m} = 2C+P-1,$ where

$C$ is the number of cycles (of length at least

$2$ ) and P is the number of paths in the meander. This statistic is

$\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{n}$ .

Matching statistic: St001673
(load all 2 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> [1] => 0
=> [1] => 0
=> [2] => 0
=> [1,1] => 0
=> [1,1] => 0
=> [2] => 0
=> [3] => 0
=> [2,1] => 1
=> [1,1,1] => 0
=> [1,2] => 1
=> [1,2] => 1
=> [1,1,1] => 0
=> [2,1] => 1
=> [3] => 0
=> [4] => 0
=> [3,1] => 1
=> [2,1,1] => 1
=> [2,2] => 0
=> [1,1,2] => 1
=> [1,1,1,1] => 0
=> [1,2,1] => 0
=> [1,3] => 1
=> [1,3] => 1
=> [1,2,1] => 0
=> [1,1,1,1] => 0
=> [1,1,2] => 1
=> [2,2] => 0
=> [2,1,1] => 1
=> [3,1] => 1
=> [4] => 0

Description

The degree of asymmetry of an integer composition. This is the number of pairs of symmetrically positioned distinct entries.

Matching statistic: St001423
(load all 3 compositions to match this statistic)

Mapping

Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St001423: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 0 => 1 => 0
=> 1 => 0 => 0
=> 00 => 01 => 0
=> 10 => 11 => 0
=> 10 => 11 => 0
=> 11 => 00 => 0
=> 000 => 001 => 0
=> 100 => 011 => 0
=> 100 => 011 => 0
=> 110 => 111 => 1
=> 100 => 011 => 0
=> 110 => 111 => 1
=> 110 => 111 => 1
=> 111 => 000 => 1
=> 0000 => 0001 => 1
=> 1000 => 0011 => 0
=> 1000 => 0011 => 0
=> 1010 => 1101 => 0
=> 1000 => 0011 => 0
=> 1100 => 0111 => 1
=> 1010 => 1101 => 0
=> 1110 => 1111 => 1
=> 1000 => 0011 => 0
=> 1010 => 1101 => 0
=> 1100 => 0111 => 1
=> 1110 => 1111 => 1
=> 1010 => 1101 => 0
=> 1110 => 1111 => 1
=> 1110 => 1111 => 1
=> 1111 => 0000 => 1

Description

The number of distinct cubes in a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words

$u$ such that

$uuu$ is a factor of the word.

Matching statistic: St000292

Mapping

Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 1 => 1 => 1 => 0
=> 1 => 1 => 1 => 0
=> 01 => 00 => 00 => 0
=> 10 => 11 => 11 => 0
=> 11 => 10 => 10 => 0
=> 11 => 10 => 10 => 0
=> 001 => 011 => 011 => 1
=> 010 => 000 => 000 => 0
=> 101 => 111 => 111 => 0
=> 101 => 111 => 111 => 0
=> 101 => 111 => 111 => 0
=> 110 => 100 => 010 => 1
=> 111 => 101 => 101 => 1
=> 111 => 101 => 101 => 1
=> 0001 => 0100 => 0100 => 1
=> 0010 => 0111 => 0111 => 1
=> 0101 => 0000 => 0000 => 0
=> 0101 => 0000 => 0000 => 0
=> 1001 => 1100 => 0110 => 1
=> 1010 => 1111 => 1111 => 0
=> 1011 => 1110 => 1110 => 0
=> 1011 => 1110 => 1110 => 0
=> 1001 => 1100 => 0110 => 1
=> 1010 => 1111 => 1111 => 0
=> 1101 => 1000 => 0010 => 1
=> 1101 => 1000 => 0010 => 1
=> 1101 => 1000 => 0010 => 1
=> 1110 => 1011 => 1011 => 1
=> 1111 => 1010 => 1100 => 0
=> 1111 => 1010 => 1100 => 0

Description

The number of ascents of a binary word.

Matching statistic: St000506
(load all 4 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [1] => [1] => [1]
 => 0
1 => [1] => [1] => [1]
 => 0
00 => [2] => [1] => [1]
 => 0
01 => [1,1] => [2] => [2]
 => 0
10 => [1,1] => [2] => [2]
 => 0
11 => [2] => [1] => [1]
 => 0
000 => [3] => [1] => [1]
 => 0
001 => [2,1] => [1,1] => [1,1]
 => 1
010 => [1,1,1] => [3] => [3]
 => 0
011 => [1,2] => [1,1] => [1,1]
 => 1
100 => [1,2] => [1,1] => [1,1]
 => 1
101 => [1,1,1] => [3] => [3]
 => 0
110 => [2,1] => [1,1] => [1,1]
 => 1
111 => [3] => [1] => [1]
 => 0
0000 => [4] => [1] => [1]
 => 0
0001 => [3,1] => [1,1] => [1,1]
 => 1
0010 => [2,1,1] => [1,2] => [2,1]
 => 1
0011 => [2,2] => [2] => [2]
 => 0
0100 => [1,1,2] => [2,1] => [2,1]
 => 1
0101 => [1,1,1,1] => [4] => [4]
 => 0
0110 => [1,2,1] => [1,1,1] => [1,1,1]
 => 0
0111 => [1,3] => [1,1] => [1,1]
 => 1
1000 => [1,3] => [1,1] => [1,1]
 => 1
1001 => [1,2,1] => [1,1,1] => [1,1,1]
 => 0
1010 => [1,1,1,1] => [4] => [4]
 => 0
1011 => [1,1,2] => [2,1] => [2,1]
 => 1
1100 => [2,2] => [2] => [2]
 => 0
1101 => [2,1,1] => [1,2] => [2,1]
 => 1
1110 => [3,1] => [1,1] => [1,1]
 => 1
1111 => [4] => [1] => [1]
 => 0

Description

The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry

$i$ such that

$i+1$ appears to the right or above

$i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.

Matching statistic: St001071

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001071: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [1] => [1] => ([],1)
 => 0
1 => [1] => [1] => ([],1)
 => 0
00 => [2] => [1] => ([],1)
 => 0
01 => [1,1] => [2] => ([],2)
 => 0
10 => [1,1] => [2] => ([],2)
 => 0
11 => [2] => [1] => ([],1)
 => 0
000 => [3] => [1] => ([],1)
 => 0
001 => [2,1] => [1,1] => ([(0,1)],2)
 => 1
010 => [1,1,1] => [3] => ([],3)
 => 0
011 => [1,2] => [1,1] => ([(0,1)],2)
 => 1
100 => [1,2] => [1,1] => ([(0,1)],2)
 => 1
101 => [1,1,1] => [3] => ([],3)
 => 0
110 => [2,1] => [1,1] => ([(0,1)],2)
 => 1
111 => [3] => [1] => ([],1)
 => 0
0000 => [4] => [1] => ([],1)
 => 0
0001 => [3,1] => [1,1] => ([(0,1)],2)
 => 1
0010 => [2,1,1] => [1,2] => ([(1,2)],3)
 => 1
0011 => [2,2] => [2] => ([],2)
 => 0
0100 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 0
0101 => [1,1,1,1] => [4] => ([],4)
 => 0
0110 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
0111 => [1,3] => [1,1] => ([(0,1)],2)
 => 1
1000 => [1,3] => [1,1] => ([(0,1)],2)
 => 1
1001 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
1010 => [1,1,1,1] => [4] => ([],4)
 => 0
1011 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 0
1100 => [2,2] => [2] => ([],2)
 => 0
1101 => [2,1,1] => [1,2] => ([(1,2)],3)
 => 1
1110 => [3,1] => [1,1] => ([(0,1)],2)
 => 1
1111 => [4] => [1] => ([],1)
 => 0

Description

The beta invariant of the graph. The beta invariant was introduced by Crapo [1] for matroids. For graphs with

$n$ vertices the beta invariant is

$\beta(G) = (-1)^{n-c}\sum_{S\subseteq E} (-1)^{|S|} (n-c(S)),$ where

$c(S)$ is the number of connected components of the subgraph of

$G$ with edge set

$S$ . For graphs with at least one edge the beta invariant equals the absolute value of the derivative of the chromatic polynomial at

$1$ . [2] The beta invariant also coincides with the coefficient of the monomial

$x$ , and also with the coefficient of the monomial

$y$ , of the Tutte polynomial.

Matching statistic: St001092

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [1] => [1,0]
 => []
 => 0
1 => [1] => [1,0]
 => []
 => 0
00 => [2] => [1,1,0,0]
 => []
 => 0
01 => [1,1] => [1,0,1,0]
 => [1]
 => 0
10 => [1,1] => [1,0,1,0]
 => [1]
 => 0
11 => [2] => [1,1,0,0]
 => []
 => 0
000 => [3] => [1,1,1,0,0,0]
 => []
 => 0
001 => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => [2,1]
 => 1
011 => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 0
100 => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 0
101 => [1,1,1] => [1,0,1,0,1,0]
 => [2,1]
 => 1
110 => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 1
111 => [3] => [1,1,1,0,0,0]
 => []
 => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
 => []
 => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
1111 => [4] => [1,1,1,1,0,0,0,0]
 => []
 => 0

Description

The number of distinct even parts of a partition. See Section 3.3.1 of [1].

Matching statistic: St001115
(load all 2 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [1] => [1,0]
 => [1] => 0
1 => [1] => [1,0]
 => [1] => 0
00 => [2] => [1,1,0,0]
 => [1,2] => 0
01 => [1,1] => [1,0,1,0]
 => [2,1] => 0
10 => [1,1] => [1,0,1,0]
 => [2,1] => 0
11 => [2] => [1,1,0,0]
 => [1,2] => 0
000 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0
001 => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 0
010 => [1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => 1
011 => [1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 1
100 => [1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => 1
110 => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 0
111 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0

Description

The number of even descents of a permutation.

Matching statistic: St001125

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [1] => [1] => [1,0]
 => 0
1 => [1] => [1] => [1,0]
 => 0
00 => [2] => [1] => [1,0]
 => 0
01 => [1,1] => [2] => [1,1,0,0]
 => 0
10 => [1,1] => [2] => [1,1,0,0]
 => 0
11 => [2] => [1] => [1,0]
 => 0
000 => [3] => [1] => [1,0]
 => 0
001 => [2,1] => [1,1] => [1,0,1,0]
 => 1
010 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 0
011 => [1,2] => [1,1] => [1,0,1,0]
 => 1
100 => [1,2] => [1,1] => [1,0,1,0]
 => 1
101 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 0
110 => [2,1] => [1,1] => [1,0,1,0]
 => 1
111 => [3] => [1] => [1,0]
 => 0
0000 => [4] => [1] => [1,0]
 => 0
0001 => [3,1] => [1,1] => [1,0,1,0]
 => 1
0010 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
 => 1
0011 => [2,2] => [2] => [1,1,0,0]
 => 0
0100 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 1
0101 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
0110 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
0111 => [1,3] => [1,1] => [1,0,1,0]
 => 1
1000 => [1,3] => [1,1] => [1,0,1,0]
 => 1
1001 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
1010 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
1011 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
 => 1
1100 => [2,2] => [2] => [1,1,0,0]
 => 0
1101 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
 => 1
1110 => [3,1] => [1,1] => [1,0,1,0]
 => 1
1111 => [4] => [1] => [1,0]
 => 0

Description

The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.

The following 161 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001271The competition number of a graph. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001556The number of inversions of the third entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St000630The length of the shortest palindromic decomposition of a binary word. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001151The number of blocks with odd minimum. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000486The number of cycles of length at least 3 of a permutation. St001557The number of inversions of the second entry of a permutation. St000402Half the size of the symmetry class of a permutation. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000886The number of permutations with the same antidiagonal sums. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001204Call 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 special CNakayama algebra. St001587Half of the largest even part of an integer partition. St000260The radius of a connected graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000941The number of characters of the symmetric group whose value on the partition is even. St001095The number of non-isomorphic posets with precisely one further covering relation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000447The number of pairs of vertices of a graph with distance 3. St001306The number of induced paths on four vertices in a graph. St001350Half of the Albertson index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St000535The rank-width of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001393The induced matching number of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001694The number of maximal dissociation sets in a graph. St001716The 1-improper chromatic number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St000379The number of Hamiltonian cycles in a graph. St000944The 3-degree of an integer partition. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001282The number of graphs with the same chromatic polynomial. St001315The dissociation number of a graph. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000478Another weight of a partition according to Alladi. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000699The toughness times the least common multiple of 1,. St000934The 2-degree of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001812The biclique partition number of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000781The number of proper colouring schemes of a Ferrers diagram. St000940The number of characters of the symmetric group whose value on the partition is zero. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. 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. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001541The Gini index of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001651The Frankl number of a lattice. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001763The Hurwitz number of an integer partition. 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer 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. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001722The number of minimal chains with small intervals between a binary word and the top element. St000455The second largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. 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. 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. St001128The exponens consonantiae of a partition. St000782The indicator function of whether a given perfect matching is an L & P matching.