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Two EC tidbits Sergi Elizalde Dartmouth College In honor of - PowerPoint PPT Presentation

Lattice paths Walks in the plane 321 -avoiding involutions Two EC tidbits Sergi Elizalde Dartmouth College In honor of Richard Stanleys 70th birthday Sergi Elizalde Two EC tidbits Lattice paths Walks in the plane 321 -avoiding


  1. 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

  2. Lattice paths Walks in the plane 321 -avoiding involutions Sergi Elizalde Two EC tidbits

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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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