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Permutohedra, Associahedra, and Beyond or Three Formulas for Volumes of Permutohedra by Alex Postnikov Massachusetts Institute of Technology June 26, 2004 on the occasion of Richard P. Stanleys Birthday 1 Permutohedron P n ( x 1 , . . . ,

  1. Permutohedra, Associahedra, and Beyond or Three Formulas for Volumes of Permutohedra by Alex Postnikov Massachusetts Institute of Technology June 26, 2004 on the occasion of Richard P. Stanley’s Birthday 1

  2. Permutohedron P n ( x 1 , . . . , x n +1 ) := ConvexHull(( x w (1) , . . . , x w ( n +1) ) | w ∈ S n +1 ) This is a convex n -dimensional polytope in H ⊂ R n +1 . Example: n = 2 (type A 2 ) ( x 1 , x 2 , x 3 ) P 2 ( x 1 , x 2 , x 3 ) = More generaly, for a Weyl group W , P W ( x ) := ConvexHull( w ( x ) | w ∈ W ). 2

  3. Question: What is the volume V n := Vol P n ? Volume form is normalized so that the volume of a parallelepiped formed by generators of the lattice Z n +1 ∩ H is 1. Question: What is the number of lattice points N n := P n ∩ Z n +1 ? We will see that V n and N n are polynomials in x 1 , . . . , x n +1 of degree n . The polynomial V n is the top homogeneous part of N n . The Ehrhart polynomial of P n is E ( t ) = N n ( tx 1 , . . . , tx n ), and V n is its top coefficient. We will give 3 totally different formulas for these polynomials. 3

  4. Special Case: P n ( n + 1 , n, . . . , 1) = ConvexHull(( w (1) , ..., w ( n + 1)) | w ∈ S n +1 ) is the most symmetric permutohedron. regular hexagon subdivided into 3 rhombi It is a zonotope = Minkowsky sum of line intervals. Well-known facts: ➠ V n ( n + 1 , . . . , 1) = ( n + 1) n − 1 is the number of trees on n + 1 labelled vertices. P n ( n + 1 , . . . , 1) can be subdivided into parallelepipeds of unit volume associated with trees. This works for any zonotope. ➠ N n ( n + 1 , . . . , 1) is the number of forests on n + 1 labelled vertices. 4

  5. First Formula Fix any distinct numbers λ 1 , . . . , λ n +1 such that λ 1 + · · · + λ n +1 = 0. ( λ w (1) x 1 + · · · + λ w ( n +1) x n +1 ) n V n ( x 1 , . . . , x n +1 ) = 1 � n ! ( λ w (1) − λ w (2) )( λ w (2) − λ w (3) ) · · · ( λ w ( n ) − λ w ( n +1) ) w ∈ S n +1 Notice that the symmetrization in RHS does not depends on λ i ’s. Use Khovansky-Puchlikov’s method: Idea of Proof ➠ Instead of just counting the number of lattice points in P , define [ P ] = sum of formal exponents e a over lattice points a ∈ P ∩ Z n . ➠ Now we can work with unbounded polytopes. For example, for a simplicial cone C , the sum [ C ] is given by a simple rational expression. ➠ Any polytope P can be explicitly presented as an alternating sum of simplicial cones: [ P ] = [ C 1 ] ± [ C 2 ] ± · · · . Applying this procedure to the permutohedron, we obtain . . . 5

  6. Let α 1 , . . . , α n be a system of simple roots for Weyl group W , and let L be the root lattice. Theorem: For a dominant weight µ , e w ( µ ) e a = � � [ P W ( µ )] := (1 − e − w ( α 1 ) ) · · · (1 − e − w ( α n ) ) w ∈ W a ∈ P W ( µ ) ∩ ( L + µ ) Compare this with Weyl’s character formula! Note: LHS is obtained from the character ch V µ of the irrep V µ by replacing all nonzero coefficients with 1. In type A , ch V µ = Schur polynomial s µ . From this expression, one can deduce the First Formula and also its gen- eralizations to other Weyl groups. 6

  7. Second Formula Let us use the coordinates y 1 , . . . , y n +1 related x 1 , . . . , x n +1 by  y 1 = − x 1      y 2 = − x 2 + x 1     y 3 = − x 3 + 2 x 2 − x 1   · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·      � n � n � n � � �  y n +1 = − x n + x n − 1 − · · · ± x 1  0 1 n and write V n = Vol P n ( x 1 , . . . , x n +1 ) as a polynomial in y 1 , . . . , y n +1 . Examples: V 1 = Vol ([( x 1 , x 2 ) , ( x 2 , x 1 )]) = x 1 − x 2 = y 2 2 + 3 y 2 y 3 + 1 V 2 = · · · = 3 y 2 2 y 2 3 7

  8. Theorem: V n ( x 1 , . . . , x n +1 ) = 1 � y | S 1 | · · · y | S n | , n ! ( S 1 ,...,S n ) where the sum is over ordered collections of subsets S 1 , . . . , S n ⊂ [ n + 1] such that either of the following equivalent conditions is satisfied: ➠ For any distinct i 1 , . . . , i k , we have | S i 1 ∪ · · · ∪ S i k | ≥ k + 1 (cf. Hall’s Marriage Theorem) ➠ For any j ∈ [ n + 1], there is a system of distinct representatives in S 1 , . . . , S n that avoids j . Thus n ! V n is a polynomial in y 2 , . . . , y n +1 with positive integer coefficients. 8

  9. This formula can be extended to generalized permutohedra a generalized permutohedron Generalized permutohedra are obtained from usual permutohedra by mov- ing faces while preserving all angles. this is also a generalized permutohedron 9

  10. Generalized Permutohedra Coordinate simplices in R n +1 : ∆ I = ConvexHull( e i | i ∈ I ), for I ⊆ [ n +1]. Let Y = { Y I } be the collection of variables Y I ≥ 0 associated with all subsets I ⊂ [ n + 1]. Define � P n ( Y ) := Y I · ∆ I (Minkowsky sum) I ⊂ [ n +1] Its combinatorial type depends only on the set of I ’s for which Y I � = 0. Examples: ➠ If Y I = y | I | , then P n ( Y ) is a usual permutohedron. ➠ If Y I � = 0 iff I is a consecutive interval, then P n ( Y ) is an associahedron. ➠ If Y I � = 0 iff I is a cyclic interval, then P n ( Y ) is a cyclohedron. ➠ If Y I � = 0 iff I is a connected set in Dynkin diagram, then P n ( Y ) is a generalized associahedron related to DeConcini-Procesi’s work. (Do not confuse with Fomin-Zelevinsky’s generalized associahedra!) ➠ If Y I � = 0 iff I is an initial interval { 1 , . . . , i } , then P n ( Y ) is the Stanley-Pitman polytope. 10

  11. Theorem: The volume of the generalized permutohedron is given by Vol P n ( Y ) = 1 � Y S 1 · · · Y S n , n ! ( S 1 ,...,S n ) where S 1 , . . . , S n satisfy the same condition. Theorem: The # of lattice points in the generalized permutohedron is P n ( Y ) ∩ Z n +1 = 1 � { Y S 1 · · · Y S n } , n ! ( S 1 ,...,S n ) �� � , where Y { a } = Y ( Y +1) · · · ( Y + a − 1) . := ( Y [ n +1] +1) { a [ n +1] } � Y { a I } Y a I I I I I � =[ n +1] This extends a formula from [Stanley-Pitman] for the volume of their polytope. In this case, the above summation is over parking functions. 11

  12. We also have a combinatorial description of face structure of generalized permutohedra in terms of nested collections of subsets in [ n + 1]. This is related to DeConcini-Procesi’s wonderful arrangements. Not enough time for this now. The most interesting part of the talk is . . . 12

  13. Third Formula Let use the coordinates z 1 , . . . , z n related to x 1 , . . . , x n +1 by z 1 = x 1 − x 2 , z 2 = x 2 − x 3 , · · · , z n = x n − x n +1 These coordinates are canonically defined for an arbitrary Weyl group. Then the permutohedron P n is the Minkowsky sum P n = z 1 ∆ 1 n + z 2 ∆ 2 n + · · · + z n ∆ nn of hypersimplices ∆ kn = P n (1 , . . . , 1 , 0 , . . . , 0) (with k 1’s). + = 13

  14. This implies z c 1 c 1 ! · · · z c n � 1 n Vol P n = A c 1 ,...,c n c n ! , c 1 ,...,c n where the sum is over c 1 , . . . , c n ≥ 0, c 1 + · · · + c n = n , and A c 1 ,...,c n = MixedVolume(∆ c 1 1 n , . . . , ∆ c n nn ) ∈ Z > 0 In particular, n ! V n is a positive integer polynomial in z 1 , . . . , z n . Let us call the integers A c 1 ,...,c n the Mixed Eulerian numbers. Examples: V 1 = 1 z 1 V 2 = 1 z 2 2 + 2 z 1 z 2 + 1 z 2 1 2 2 V 3 = 1 z 3 3! + 2 z 2 2 + 4 z 3 3! + 3 z 2 z 2 2 z 2 + 4 z 1 2 z 3 + 6 z 1 z 2 z 3 + 1 1 2 1 +4 z 2 z 2 z 2 2 + 1 z 3 2 z 3 + 3 z 1 2 + 2 z 2 2 3 3 3 3! (The mixed Eulerian numbers are marked in red.) 14

  15. Properties of Mixed Eulerian numbers: ➠ A c 1 ,...,c n are positive integers defined for c 1 , . . . , c n ≥ 0, c 1 + · · · + c n = n . c 1 ! ··· c n ! A c 1 ,...,c n = ( n + 1) n − 1 . 1 ➠ � ➠ A 0 ,..., 0 ,n, 0 ,..., 0 ( n is in k -th position) is the usual Eulerian number A kn = # permutations in S n with k descents = n ! Vol ∆ kn . ➠ A 1 ,..., 1 = n ! � n � ➠ A k, 0 ,..., 0 ,n − k = k ➠ A c 1 ,...,c n = 1 c 1 2 c 2 · · · n c n if c 1 + · · · + c i ≥ i , for i = 1 , . . . , n . � 2 n 1 � There are exactly C n = such sequences ( c 1 , . . . , c n ). n +1 n When I showed these numbers to Richard Stanley, he conjectured that ➠ � A c 1 ,...,c n = n ! C n . Moreover, he conjectured that . . . 15

  16. One can subdivide all sequences ( c 1 , . . . , c n ) into C n classes such that the sum of mixed Eulerian numbers for each class is n !. For example, A 1 ,..., 1 = n ! and A n, 0 ,..., 0 + A 0 ,n, 0 ,..., 0 + A 0 , 0 ,n,..., 0 + · · · + A 0 ,..., 0 ,n = n !, because the sum of Eulerian numbers � k A kn is n !. Let us write ( c 1 , . . . , c n ) ∼ ( c ′ 1 , . . . , c ′ n ) iff ( c 1 , . . . , c n , 0) is a cyclic shift of ( c ′ 1 , . . . , c ′ n , 0). Stanley conjectured that, for fixed ( c 1 , . . . , c n ), we have � n = n ! A c ′ 1 ,...,c ′ ( c ′ n ) ∼ ( c 1 ,...,c n ) 1 ,...,c ′ Exercise: Check that there are exactly C n equivalence classes of sequences. Every equivalence class contains exactly one sequence ( c 1 , . . . , c n ) such that c 1 + · · · + c i ≥ i , for i = 1 , . . . , n . (For this sequence, A c 1 ,...,c n = 1 c 1 · · · n c n .) These conjectures follow from . . . 16

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