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Your data matches 627 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001414
(load all 33 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

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: St000089
(load all 3 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
St000089: 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

Description

The absolute variation of a composition.

Matching statistic: St001095

Mapping

Mp00262: Binary words —poset of factors⟶ Posets
St001095: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => ([(0,1)],2)
 => 0
1 => ([(0,1)],2)
 => 0
00 => ([(0,2),(2,1)],3)
 => 0
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
11 => ([(0,2),(2,1)],3)
 => 0
000 => ([(0,3),(2,1),(3,2)],4)
 => 0
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 0
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 0
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
111 => ([(0,3),(2,1),(3,2)],4)
 => 0

Description

The number of non-isomorphic posets with precisely one further covering relation.

Matching statistic: St001263
(load all 4 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

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 3 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

Description

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

Matching statistic: St000277
(load all 3 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
St000277: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Matching statistic: St000767
(load all 3 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
St000767: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of runs in an integer composition. Writing the composition as

$c_1^{e_1} \dots c_\ell^{e_\ell}$ , where

$c_i \neq c_{i+1}$ for all

$i$ , the number of runs is

$\ell$ , see [def.2.8, 1]. It turns out that the total number of runs in all compositions of

$n$ equals the total number of odd parts in all these compositions, see [1].

Matching statistic: St000820
(load all 3 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
St000820: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of compositions obtained by rotating the composition.

Matching statistic: St000847
(load all 12 compositions to match this statistic)

Mapping

Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000847: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 1 => 1 = 0 + 1
=> 1 => 1 = 0 + 1
=> 01 => 1 = 0 + 1
=> 10 => 1 = 0 + 1
=> 11 => 1 = 0 + 1
=> 11 => 1 = 0 + 1
=> 001 => 1 = 0 + 1
=> 010 => 2 = 1 + 1
=> 101 => 2 = 1 + 1
=> 101 => 2 = 1 + 1
=> 101 => 2 = 1 + 1
=> 110 => 1 = 0 + 1
=> 111 => 1 = 0 + 1
=> 111 => 1 = 0 + 1

Description

The number of standard Young tableaux whose descent set is the binary word. A descent in a standard Young tableau is an entry

$i$ such that

$i+1$ appears in a lower row in English notation. For example, the tableaux

$[[1,2,4],[3]]$ and

$[[1,2],[3,4]]$ are those with descent set

$\{2\}$ , corresponding to the binary word

$010$ .

Matching statistic: St000903
(load all 3 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of different parts of an integer composition.

The following 617 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000008The major index of the composition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000447The number of pairs of vertices of a graph with distance 3. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. 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. St000921The number of internal inversions of a binary word. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001214The aft of an integer partition. 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. St001423The number of distinct cubes in a binary word. St001521Half the total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001730The number of times the path corresponding to a binary word crosses the base line. St001742The difference of the maximal and the minimal degree in a graph. St001777The number of weak descents in an integer composition. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001931The weak major index of an integer composition regarded as a word. St000003The number of standard Young tableaux of the partition. St000159The number of distinct parts of the integer partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000388The number of orbits of vertices of a graph under automorphisms. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000535The rank-width of a graph. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. St001352The number of internal nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001432The order dimension of the partition. St001595The number of standard Young tableaux of the skew partition. St001675The number of parts equal to the part in the reversed composition. St001694The number of maximal dissociation sets in a graph. St001716The 1-improper chromatic number of a graph. St001734The lettericity 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. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001261The Castelnuovo-Mumford regularity of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000002The number of occurrences of the pattern 123 in a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000023The number of inner peaks of a permutation. St000053The number of valleys of the Dyck path. St000057The Shynar inversion number of a standard tableau. St000081The number of edges of a graph. St000090The variation of a composition. St000091The descent variation of a composition. St000095The number of triangles of a graph. St000117The number of centered tunnels of a Dyck path. St000120The number of left tunnels of a Dyck path. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000171The degree of the graph. St000185The weighted size of a partition. St000204The number of internal nodes of a binary tree. 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. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000222The number of alignments in the permutation. St000242The number of indices that are not cyclical small weak excedances. St000257The number of distinct parts of a partition that occur at least twice. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000272The treewidth of a graph. St000274The number of perfect matchings of a graph. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. St000387The matching number of a graph. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000534The number of 2-rises of a permutation. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000628The balance of a binary word. St000632The jump number of the poset. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000649The number of 3-excedences of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000663The number of right floats of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000691The number of changes of a binary word. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000872The number of very big descents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000944The 3-degree of an integer partition. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001071The beta invariant of the graph. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001082The number of boxed occurrences of 123 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001115The number of even descents of a permutation. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001161The major index north count of a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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

$eA$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of 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. 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001270The bandwidth of a graph. 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. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the 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. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001341The number of edges in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001362The normalized Knill dimension of a graph. St001395The number of strictly unfriendly partitions of a graph. St001397Number of pairs of incomparable elements in a finite poset. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001479The number of bridges of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001512The minimum rank of a graph. St001520The number of strict 3-descents. St001524The degree of symmetry of a binary word. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001556The number of inversions of the third entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001638The book thickness of a graph. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001644The dimension of a 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. St001649The length of a longest trail in a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001712The number of natural descents of a standard Young tableau. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001718The number of non-empty open intervals in a poset. St001727The number of invisible inversions of a permutation. St001736The total number of cycles in a graph. St001743The discrepancy of a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001783The number of odd automorphisms of a graph. St001792The arboricity of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001856The number of edges in the reduced word graph of a permutation. St001869The maximum cut size of a graph. St001910The height of the middle non-run of a Dyck path. St001911A descent variant minus the number of inversions. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St000001The number of reduced words for a permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000015The number of peaks of a Dyck path. St000047The number of standard immaculate tableaux of a given shape. St000048The multinomial of the parts of a partition. St000071The number of maximal chains in a poset. St000075The orbit size of a standard tableau under promotion. St000078The number of alternating sign matrices whose left key is the permutation. St000079The number of alternating sign matrices for a given Dyck path. St000086The number of subgraphs. St000092The number of outer peaks of a permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000099The number of valleys of a permutation, including the boundary. St000160The multiplicity of the smallest part of a partition. St000172The Grundy number of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000298The order dimension or Dushnik-Miller dimension of a poset. St000299The number of nonisomorphic vertex-induced subtrees. St000307The number of rowmotion orbits of a poset. St000321The number of integer partitions of n that are dominated by an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000343The number of spanning subgraphs of a graph. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000363The number of minimal vertex covers of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000413The number of ordered trees with the same underlying unordered tree. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000482The (zero)-forcing number of a graph. St000522The number of 1-protected nodes of a rooted tree. St000527The width of the poset. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000543The size of the conjugacy class of a binary word. St000548The number of different non-empty partial sums of an integer partition. St000617The number of global maxima of a Dyck path. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000657The smallest part of an integer composition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000675The number of centered multitunnels of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000722The number of different neighbourhoods in a graph. St000753The Grundy value for the game of Kayles on 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. St000764The number of strong records in an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000789The number of crossing-similar perfect matchings of a perfect matching. St000808The number of up steps of the associated bargraph. 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. St000816The number of standard composition tableaux of the composition. St000822The Hadwiger number of the graph. St000829The Ulam distance of a permutation to the identity permutation. St000862The number of parts of the shifted shape of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000909The number of maximal chains of maximal size in a poset. St000913The number of ways to refine the partition into singletons. St000920The logarithmic height of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000958The number of Bruhat factorizations of a permutation. St000959The number of strong Bruhat factorizations of a permutation. St000972The composition number of a graph. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001052The length of the exterior of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001093The detour number of a graph. St001096The size of the overlap set of a permutation. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001151The number of blocks with odd minimum. St001162The minimum jump of a permutation. 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. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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. 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: St001220The width of a permutation. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001268The size of the largest ordinal summand in the poset. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001282The number of graphs with the same chromatic polynomial. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001372The length of a longest cyclic run of ones of a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001399The distinguishing number of a poset. St001415The length of the longest palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001424The number of distinct squares in a binary word. St001471The magnitude of a Dyck path. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001486The number of corners of the ribbon associated with an integer composition. St001489The maximum of the number of descents and the number of inverse descents. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001494The Alon-Tarsi number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001517The length of a longest pair of twins in a permutation. St001530The depth of a Dyck path. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001569The maximal modular displacement of a permutation. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001667The maximal size of a pair of weak twins for a permutation. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001779The order of promotion on the set of linear extensions of a poset. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001801Half the number of preimage-image pairs of different parity in a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001874Lusztig's a-function for the symmetric group. St001883The mutual visibility number of a graph. St001884The number of borders of a binary word. St001917The order of toric promotion on the set of labellings of a graph. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001963The tree-depth of a graph. St000824The sum of the number of descents and the number of recoils of a permutation. St000842The breadth of a permutation. 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. 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

$eA$ . St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001315The dissociation number of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001566The length of the longest arithmetic progression in a permutation. 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. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001570The minimal number of edges to add to make a graph Hamiltonian. St000353The number of inner valleys of a permutation. St000369The dinv deficit of a Dyck path. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000941The number of characters of the symmetric group whose value on the partition is even. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001388The number of non-attacking neighbors of a permutation. St001557The number of inversions of the second entry of a permutation. St001731The factorization defect of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000060The greater neighbor of the maximum. St000100The number of linear extensions of a poset. St000402Half the size of the symmetry class of a permutation. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000530The number of permutations with the same descent word as the given permutation. St000568The hook number of a binary tree. St000633The size of the automorphism group of a poset. St000744The length of the path to the largest entry in a standard Young tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000886The number of permutations with the same antidiagonal sums. St000910The number of maximal chains of minimal length in a poset. St000988The orbit size of a permutation under Foata's bijection. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001246The maximal difference between two consecutive entries of a permutation. St001668The number of points of the poset minus the width of the poset. St000064The number of one-box pattern of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000717The number of ordinal summands of a poset. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000906The length of the shortest maximal chain in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001175The size of a partition minus the hook length of the base cell. 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. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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. St000699The toughness times the least common multiple of 1,. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. 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. 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. St001280The number of parts of an integer partition that are at least two. St001281The normalized isoperimetric number of a graph. 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. St001587Half of the largest even part of an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St000934The 2-degree of an integer partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001541The Gini index of an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000456The monochromatic index of a connected graph. St000762The sum of the positions of the weak records of an integer composition. St001118The acyclic chromatic index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000782The indicator function of whether a given perfect matching is an L & P matching.