UnicellularChromatics
Chromatic symmetric functions and LLT polynomials of unit interval graphs and area sequences, orientations and their statistics.
Load with Needs["UnicellularChromatics`"].
Background on symmetricfunctions.com: Chromatic quasisymmetric functions, Chromatic symmetric functions in the elementary basis, Unicellular LLT polynomials and twin manifolds.
Functions and symbols
AcyclicAscents
AcyclicAscents[lambda, coloring] or AcyclicAscents[orientation] counts ascending edges.
AlternatingChains
AlternatingChains[lambda, coloring, i] returns the alternating chains using colors i and i+1.
AreaConjugate
AreaConjugate[area] returns the conjugate of a non-circular 0-first area list.
AreaDinv
AreaDinv[area] returns dinv for a 0-first area list.
AreaLists
AreaLists[size, All->False, Circular->True, Width->-1] returns all area lists of size n.
AreaRowPermutation
AreaRowPermutation[area] returns the row-to-column permutation for a 0-first area list.
AreaToDyckWord
AreaToDyckWord[area, corners] converts a 0-first area list to its Dyck word.
AreaToTopBounceShape
AreaToTopBounceShape[area] returns the top-starting bounce shape for a 0-first area list.
AthanasiadisS
AthanasiadisS[lambda] returns the Athanasiadis partial sums.
AthanasiadisUnimodalSets
AthanasiadisUnimodalSets[lambda] returns lambda-unimodal subsets of [n-1].
AttackingPoset
AttackingPoset[lambda] returns the attacking poset edges of a triangular shape.
AttackingVertices
AttackingVertices[lambda, coloring] returns monochromatic incomparability edges.
AyclicAreaListQ
AyclicAreaListQ[area] tests whether an area list has minimum zero.
BounceEndpoint
BounceEndpoint[word, y] returns the endpoint row reached by the bounce path starting at row y for a Schroeder word.
BounceLengths
BounceLengths[lambda] gives the lengths of bounce triangles, from top to bottom.
BounceList
BounceList[word, y] returns the list of x-coordinates visited by the bounce path starting at row y for a Schroeder word.
ChromaticSymmetric
ChromaticSymmetric[area,q] returns the chromatic symmetric polynomial associated with given area sequence. One can also pass a graph object as argument.
Background: Chromatic quasisymmetric functions
ChromaticSymmetricColorings
ChromaticSymmetricColorings[lambda, allowAttacking, maxColor] returns colorings of a shape.
Background: Chromatic quasisymmetric functions
ChromaticSymmetricPolynomial
ChromaticSymmetricPolynomial[lambda, x, q, n] returns the chromatic symmetric polynomial in variables x.
Background: Chromatic quasisymmetric functions
Circular
Option for AreaLists
ColorOrientation
ColorOrientation[lambda, coloring] gives the orientation induced by a coloring.
DiagramRookPlacements
DiagramRookPlacements[diagram, n] returns all non-attacking placements of n rooks in a diagram.
DinvFromAreaSeq
DinvFromAreaSeq[area] returns dinv for a 0-first Catalan area sequence.
EdgesHRVRule
EdgesHRVRule[edges] returns replacement rules assigning each vertex the list consisting of that vertex and all vertices reachable from it by ascending edge paths.
GasharovOpPTableauQ
GasharovOpPTableauQ[lambda, coloring] tests the opposite P-tableau condition.
GasharovOpPTableaux
GasharovOpPTableaux[lambda] returns all opposite P-tableau colorings.
GasharovPTableauQ
GasharovPTableauQ[lambda, coloring] tests the P-tableau condition.
GasharovPTableaux
GasharovPTableaux[lambda] returns all P-tableau colorings.
GraphAreaLists
GraphAreaLists[n, opts] returns area lists of unit interval graphs on n vertices, starting with 0 as in CatalanObjects (UnitIntervalEdges accepts them). Options: Circular -> True (default) also includes circular (cylindric) area lists; Width -> w bounds the entries by w - 1 (default n); All -> False (default) keeps one representative per rotation class, All -> True returns all.
GraphAttackingEdges
GraphAttackingEdges[edges, coloring] returns monochromatic edges.
GraphChromaticLLTPolynomial
GraphChromaticLLTPolynomial[edges, n, x, q] returns the graph LLT polynomial.
Background: Unicellular LLT polynomials and twin manifolds
GraphChromaticLLTPolynomialAttacking
GraphChromaticLLTPolynomialAttacking[equalEdges, statisticEdges, n, x, q] returns an LLT polynomial with forced attacking edges.
Background: Unicellular LLT polynomials and twin manifolds
GraphChromaticSymmetricColorings
GraphChromaticSymmetricColorings[edges, n, allowAttacking, maxColor] returns graph colorings.
Background: Chromatic quasisymmetric functions
GraphChromaticSymmetricPolynomial
GraphChromaticSymmetricPolynomial[edges, n, x, q] returns the graph chromatic symmetric polynomial.
Background: Chromatic quasisymmetric functions
GraphColoringAscents
GraphColoringAscents[edges, coloring] returns the number of edges whose color increases along the ordered edge.
GraphColoringInversions
GraphColoringInversions[edges, coloring] counts descending edges.
GraphColoringMonochromaticEdges
GraphColoringMonochromaticEdges[edges, coloring] returns the number of edges whose endpoints have equal colors.
GraphColoringOrientation
GraphColoringOrientation[edges, coloring] returns the induced orientation.
GraphOrientationAscents
GraphOrientationAscents[edges, orientation] counts correctly oriented edges.
GraphOrientationHalfSinks
GraphOrientationHalfSinks[edges, orientation, n] returns half-sinks.
GraphOrientationHalfSources
GraphOrientationHalfSources[edges, orientation, n] returns half-sources.
GraphOrientationIntersection
GraphOrientationIntersection[edges, orientation] returns equally oriented edges.
GraphOrientationInversions
GraphOrientationInversions[edges, orientation] counts oppositely oriented edges.
GraphOrientationSinks
GraphOrientationSinks[orientation, n] returns the sinks.
GraphOrientationSources
GraphOrientationSources[orientation, n] returns the sources.
HomogeneousGraphChromaticSymmetricPolynomial
HomogeneousGraphChromaticSymmetricPolynomial[edges, n, x, q, t] returns the homogeneous graph chromatic polynomial.
Background: Chromatic quasisymmetric functions
HomogeneousGraphLLTPolynomial
HomogeneousGraphLLTPolynomial[edges, n, x, q, t] returns the homogeneous graph LLT polynomial.
Background: Unicellular LLT polynomials and twin manifolds
IncomparabilityGraph
IncomparabilityGraph[lambda] returns the incomparability graph edges of a triangular shape.
InnerCorners
InnerCorners[area] returns the inner corners of a 0-first area list.
LLTOrientationForest
LLTOrientationForest[area, orientation] returns the orientation forest map.
Background: Unicellular LLT polynomials and twin manifolds
LLTOrientationLowestReachableVertex
LLTOrientationLowestReachableVertex[area, orientation] returns the lowest reachable vertex of each vertex.
Background: Unicellular LLT polynomials and twin manifolds
LLTOrientationShape
LLTOrientationShape[area, orientation] returns the orientation shape.
Background: Unicellular LLT polynomials and twin manifolds
LLTOrientationVertexPartition
LLTOrientationVertexPartition[area, orientation] returns the induced vertex partition.
Background: Unicellular LLT polynomials and twin manifolds
MajFromAreaSeq
MajFromAreaSeq[area] returns the major index of a 0-first area sequence.
NoAscendingCycleOrientations
NoAscendingCycleOrientations[edges] returns orientations with no ascending directed cycle.
OrientationPlot
OrientationPlot[area, orientation] returns a Graphics of an oriented unit interval graph.
OuterCorners
OuterCorners[area] returns the outer corners of a 0-first area list.
PArray
PArray[coloring] returns the P-array associated with a coloring.
PArrayPlot
PArrayPlot[lambda, coloring] returns a Graphics of a coloring as a P-array.
PartitionRookPlacements
PartitionRookPlacements[lambda] returns permutations fitting in a partition diagram.
PathShapes
PathShapes[n] gives all partitions that fit inside the size n triangle.
SchroederAcyclicOrientations
SchroederAcyclicOrientations[word] returns all acyclic orientations of the unit interval graph encoded by a Schroeder word, respecting its strict edges.
SchroederColoringAscents
SchroederColoringAscents[word, coloring] returns the number of ascents of a coloring on the graph encoded by a Schroeder word.
SchroederColorings
SchroederColorings[word, ncols, Partition -> True] returns colorings of the graph encoded by a Schroeder word satisfying its strict edges. ncols defaults to 0; with Partition -> False, colors range from 1 to ncols.
SchroederLLTSymmetric
SchroederLLTSymmetric[word, q] returns the LLT symmetric polynomial associated with a Schroeder word and parameter q.
Background: Unicellular LLT polynomials and twin manifolds
SchroederOrientations
SchroederOrientations[word] returns all orientations of the unit interval graph encoded by a Schroeder word, respecting its strict edges.
SchroederPlot
SchroederPlot[word] returns a plot of the area and strict edges encoded by a Schroeder word.
SchroederWordStrictEdges
SchroederWordStrictEdges[word] returns the strict edge list extracted from a Schroeder word.
SchroederWordToArea
SchroederWordToArea[word] converts a Schroeder word, given as a string or list using -/n, +/e, and 0/d, to {area, strictEdges}.
SingleCelledLLTPolynomial
SingleCelledLLTPolynomial[lambda, x, q, n] returns the single-celled LLT polynomial.
Background: Unicellular LLT polynomials and twin manifolds
SouthEastDiagram
SouthEastDiagram[permutation] returns the south-east diagram.
SouthWestDiagram
SouthWestDiagram[permutation] returns the south-west diagram.
StripSizesToEdges
StripSizesToEdges[sizes] returns {area, strictEdges}, using a 0-first area list.
UnicellularLLTSymmetric
UnicellularLLTSymmetric[area,q] returns the
unicellular LLT polynomial associated with given area sequence.
Background: Unicellular LLT polynomials and twin manifolds
UnicellularLLTSymmetricSchur
UnicellularLLTSymmetricSchur[area, q, ss] returns the LLT polynomial for an area sequence in the basis supplied by the function ss; q defaults to 1.
Background: Unicellular LLT polynomials and twin manifolds
UnitIntervalData
UnitIntervalData[area, coloring] returns the unit-interval representation of a coloring.
UnitIntervalEdges
UnitIntervalEdges[area] returns the edges of the unit interval graph.
UnitIntervalPlot
UnitIntervalPlot[area, coloring] returns a Graphics of a coloring in unit interval order.
ValleyEdges
ValleyEdges[edges, n] returns the edges that are valleys in the diagram.
VerticalStripLLTColorings
VerticalStripLLTColorings[sizes] returns colorings satisfying the strip strictness conditions.
Background: Unicellular LLT polynomials and twin manifolds
VerticalStripLLTPolynomial
VerticalStripLLTPolynomial[sizes, x, q] returns a vertical-strip LLT polynomial.
Background: Unicellular LLT polynomials and twin manifolds
Width
Option for AreaLists