Your data matches 167 different statistics following compositions of up to 3 maps.
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Mp00306: Posets rowmotion cycle typeInteger partitions
St001571: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 2
([],2)
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> 3
([],3)
=> [2,2,2,2]
=> 2
([(1,2)],3)
=> [6]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> 2
Description
The Cartan determinant of the integer partition. Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$. Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Matching statistic: St001392
Mp00306: Posets rowmotion cycle typeInteger partitions
St001392: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 1 = 2 - 1
([],2)
=> [2,2]
=> 1 = 2 - 1
([(0,1)],2)
=> [3]
=> 2 = 3 - 1
([],3)
=> [2,2,2,2]
=> 1 = 2 - 1
([(1,2)],3)
=> [6]
=> 5 = 6 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [4]
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> 1 = 2 - 1
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
Description
The position of the first return of a Dyck path.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1]
=> 2
([],2)
=> [2,2]
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 3
([],3)
=> [2,2,2,2]
=> [4,4]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 2
Description
The multiplicity of the largest part of an integer partition.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001498: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
Description
The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1]
=> 2
([],2)
=> [2,2]
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 3
([],3)
=> [2,2,2,2]
=> [4,4]
=> 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 2
Description
The largest multiplicity of a part in an integer partition.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([],3)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7 = 6 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
Description
The position of the first down step of a Dyck path.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> 1 = 2 - 1
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
The following 157 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001091The number of parts in an integer partition whose next smaller part has the same size. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. 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. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. 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. 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000011The number of touch points (or returns) of a Dyck path. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000444The length of the maximal rise of a Dyck path. St000617The number of global maxima of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000877The depth of the binary word interpreted as a path. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000160The multiplicity of the smallest part of a partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000376The bounce deficit of a Dyck path. St000674The number of hills of a Dyck path. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001312Number of parabolic noncrossing partitions indexed by the composition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000369The dinv deficit of a Dyck path. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000462The major index minus the number of excedences of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000120The number of left tunnels of a Dyck path. St000335The difference of lower and upper interactions. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. 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. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001368The number of vertices of maximal degree in a graph. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000331The number of upper interactions of a Dyck path. St000968We 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 $c_{n−1}$. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000344The number of strongly connected outdegree sequences of a graph. St000454The largest eigenvalue of a graph if it is integral. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001140Number of indecomposable modules with projective and injective dimension at least two 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule 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 $eA$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000100The number of linear extensions of a poset. St000180The number of chains of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000633The size of the automorphism group of a poset. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001330The hat guessing number of a graph. St001545The second Elser number of a connected graph. St001808The box weight or horizontal decoration of a Dyck path. St001909The number of interval-closed sets of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000289The decimal representation of a binary word. St000301The number of facets of the stable set polytope of a graph. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000656The number of cuts of a poset. St000675The number of centered multitunnels of a Dyck path. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000981The length of the longest zigzag subpath. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001361The number of lattice paths of the same length that stay weakly above a Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001885The number of binary words with the same proper border set. St000460The hook length of the last cell along the main diagonal of an integer partition. St000464The Schultz index of a connected graph. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000997The even-odd crank of an integer partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000741The Colin de Verdière graph invariant. St001060The distinguishing index of a graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000937The number of positive values of the symmetric group character corresponding to the partition. St001098The 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 vertex labelled trees. St001118The acyclic chromatic index of a graph. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.