There are 33 maps with codomain Dyck paths in the database:
(and possibly some waiting for verification)
injective
graded
bijective
surjective
involutive
idempotent
Number of maps: 329
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Alternating sign matricesAlternating sign matricesDyck pathsPerfect matchingsPermutationsSemistandard tableauxBinary treesBinary treesGraphsOrdered treesPosetsBinary wordsBinary wordsInteger compositionsCoresInteger partitionsDecorated permutationsDyck pathsSet partitionsSkew partitionsStandard tableauxFinite Cartan typesGelfand-Tsetlin patternsGelfand-Tsetlin patternsGraphsLatticesInteger compositionsInteger partitionsLatticesOrdered set partitionsOrdered set partitionsOrdered treesParking functionsParking functionsPerfect matchingsPermutationsSigned permutationsPlane partitionsPlane partitionsPosetsSemistandard tableauxSet partitionsSigned permutationsSkew partitionsStandard tableaux
Alternating sign matricesBinary treesDyck pathsInteger compositionsInteger partitionsOrdered treesParking functionsPerfect matchingsPermutationsDyck pathsto Dyck pathto Dyck path: up step, left tree, down step, right treepruning number to logarithmic heightto Tamari-corresponding Dyck pathdecomposition reversepeeling mapswap returns and last descentLalanne-Kreweras involutionCori-Le Borgne involutionElizalde-Deutsch bijectionBarnabei-Castronuovo involutionAdin-Bagno-Roichman transformationswitch returns and last double risepromotioninverse promotionprime Dyck pathpeaks-to-valleysDelest-Viennot-inversereflect parallelogram polyominoDelest-ViennotreverseKnuth-Krattenthalerzeta mapinverse zeta mapinverse Kreweras complementbounce pathbounce pathparallelogram polyominoto Dyck pathto Dyck pathto Dyck pathto Dyck pathleft-to-right-maxima to Dyck path