Lattice paths Walks in the plane 321 -avoiding involutions Two EC tidbits Sergi Elizalde Dartmouth College In honor of Richard Stanley’s 70th birthday Sergi Elizalde Two EC tidbits
Lattice paths Walks in the plane 321 -avoiding involutions Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Tidbit 1 A bijection for pairs of non-crossing lattice paths Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Tidbit 1 A bijection for pairs of non-crossing lattice paths Stanley #70 Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Grand Dyck paths and Dyck path prefixes We consider two kinds of lattice paths with steps U = ( 1 , 1 ) and D = ( 1 , − 1 ) starting at the origin. Grand Dyck paths end on the x -axis (or at height 1 for paths of odd length) : G n = set of Grand Dyck paths of length n . Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Grand Dyck paths and Dyck path prefixes We consider two kinds of lattice paths with steps U = ( 1 , 1 ) and D = ( 1 , − 1 ) starting at the origin. Grand Dyck paths end on the x -axis (or at height 1 for paths of odd length) : G n = set of Grand Dyck paths of length n . Dyck path prefixes never go below x -axis, but can end at any height: P n = set of Dyck path prefixes of length n . Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Grand Dyck paths and Dyck path prefixes We consider two kinds of lattice paths with steps U = ( 1 , 1 ) and D = ( 1 , − 1 ) starting at the origin. Grand Dyck paths end on the x -axis (or at height 1 for paths of odd length) : G n = set of Grand Dyck paths of length n . � n � Trivial: |G n | = . ⌊ n 2 ⌋ Dyck path prefixes never go below x -axis, but can end at any height: P n = set of Dyck path prefixes of length n . Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Grand Dyck paths and Dyck path prefixes We consider two kinds of lattice paths with steps U = ( 1 , 1 ) and D = ( 1 , − 1 ) starting at the origin. Grand Dyck paths end on the x -axis (or at height 1 for paths of odd length) : G n = set of Grand Dyck paths of length n . � n � Trivial: |G n | = . ⌊ n 2 ⌋ Dyck path prefixes never go below x -axis, but can end at any height: P n = set of Dyck path prefixes of length n . � n � Not so trivial: |P n | = . ⌊ n 2 ⌋ Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions A classical bijection ξ : P n → G n P n Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions A classical bijection ξ : P n → G n P n ◮ Match U s and D s that “face" each other. Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions A classical bijection ξ : P n → G n P n ◮ Match U s and D s that “face" each other. ◮ Among the unmatched �→ ξ G n steps (which are all U s), change the lefmost half of them into D steps. Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions A classical bijection ξ : P n → G n P n ◮ Match U s and D s that “face" each other. ◮ Among the unmatched �→ ξ G n steps (which are all U s), change the lefmost half of them into D steps. To reverse, simply change unmatched D s into U s. Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions k -tuples of non-crossing paths For lattice paths P and Q , write Q ≤ P if Q is weakly below P . ( P 1 , . . . , P k ) is a k -tuple of nested paths if P k ≤ · · · ≤ P 1 . Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions k -tuples of non-crossing paths For lattice paths P and Q , write Q ≤ P if Q is weakly below P . ( P 1 , . . . , P k ) is a k -tuple of nested paths if P k ≤ · · · ≤ P 1 . G ( k ) = k -tuples of nested paths in G n n P ( k ) = k -tuples of nested paths in P n n Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions k -tuples of non-crossing paths For lattice paths P and Q , write Q ≤ P if Q is weakly below P . ( P 1 , . . . , P k ) is a k -tuple of nested paths if P k ≤ · · · ≤ P 1 . G ( k ) = k -tuples of nested paths in G n n Gessel–Viennot, MacMahon: �� k �� n |G ( k ) n | = det ⌊ n 2 ⌋ − i + j i , j = 1 ⌈ n 2 ⌉ ⌊ n 2 ⌋ k i + j + l − 1 � � � = i + j + l − 2 i = 1 j = 1 l = 1 P ( k ) = k -tuples of nested paths in P n n Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions k -tuples of non-crossing paths For lattice paths P and Q , write Q ≤ P if Q is weakly below P . ( P 1 , . . . , P k ) is a k -tuple of nested paths if P k ≤ · · · ≤ P 1 . G ( k ) = k -tuples of nested paths in G n n Gessel–Viennot, MacMahon: �� k �� n |G ( k ) n | = det ⌊ n 2 ⌋ − i + j i , j = 1 ⌈ n 2 ⌉ ⌊ n 2 ⌋ k i + j + l − 1 � � � = i + j + l − 2 i = 1 j = 1 l = 1 P ( k ) = k -tuples of nested paths in P n n |P ( k ) n | = ? Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Richard Stanley to the rescue Computing the first few terms, it seems that |G ( k ) n | = |P ( k ) n | . I asked Richard if this was known... Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Richard Stanley to the rescue Computing the first few terms, it seems that |G ( k ) n | = |P ( k ) n | . I asked Richard if this was known... Yes! [EC1, Exercise 3.47(f)] Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Richard Stanley to the rescue Computing the first few terms, it seems that |G ( k ) n | = |P ( k ) n | . I asked Richard if this was known... Yes! [EC1, Exercise 3.47(f)] Prove that the following posets have the same order polynomial: ◮ q × p (product of two chains), ◮ pairs { ( i , j ) : 1 ≤ i ≤ j ≤ p + q − i , 1 ≤ i ≤ q } ordered by ( i , j ) ≤ ( i ′ , j ′ ) if i ≤ i ′ and j ≤ j ′ . Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Richard Stanley to the rescue Computing the first few terms, it seems that |G ( k ) n | = |P ( k ) n | . I asked Richard if this was known... Yes! [EC1, Exercise 3.47(f)] Prove that the following posets have the same order polynomial: ◮ q × p (product of two chains), ◮ pairs { ( i , j ) : 1 ≤ i ≤ j ≤ p + q − i , 1 ≤ i ≤ q } ordered by ( i , j ) ≤ ( i ′ , j ′ ) if i ≤ i ′ and j ≤ j ′ . For p = q , this is equivalent to |G ( k ) n | = |P ( k ) n | . Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Richard Stanley to the rescue This was proved by Robert Proctor in the following form: Theorem (Proctor ’83) # shifted plane partitions # plane partitions inside inside shifted shape rectangle shape ( p q ) = [ p + q − 1 , p + q − 3 , . . . , p − q + 1 ] with entries ≤ k with entries ≤ k Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions Richard Stanley to the rescue This was proved by Robert Proctor in the following form: Theorem (Proctor ’83) # shifted plane partitions # plane partitions inside inside shifted shape rectangle shape ( p q ) = [ p + q − 1 , p + q − 3 , . . . , p − q + 1 ] with entries ≤ k with entries ≤ k Proctor’s proof uses representations of semisimple Lie algebras, and it is not bijective. Sergi Elizalde Two EC tidbits
Lattice paths Grand Dyck paths and Dyck path prefixes Walks in the plane A bijection for pairs of paths 321 -avoiding involutions A bijective proof for k = 2 E. ’14: Explicit bijection G ( 2 ) → P ( 2 ) n . n G ( 2 ) n P Q Sergi Elizalde Two EC tidbits
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