Flag Descents and P-Partitions Ira M. Gessel Department of Mathematics Brandeis University 2010 Joint Mathematics Meeting Special Session on Permutations San Francisco, CA January 16, 2010
Ehrhart polynomials Let K be convex polytope in R n with lattice points as vertices. Then for any positive integer m , #( mK ∩ Z n ) is a polynomial L K ( m ) in m , called the Ehrhart polynomial of K .
Ehrhart polynomials Let K be convex polytope in R n with lattice points as vertices. Then for any positive integer m , #( mK ∩ Z n ) is a polynomial L K ( m ) in m , called the Ehrhart polynomial of K . For example, if K is the unit square in R 2 (0 , 1) (1 , 1) (0 , 0) (1 , 0) then L K ( m ) = ( m + 1 ) 2 .
If the interior of K is nonempty then L K ( m ) has degree n , so ∞ A ( t ) L K ( m ) t m = � ( 1 − t ) n + 1 m = 0 for some polynomial A ( t ) of degree at most n . It is known that A ( t ) has positive coefficients.
If the interior of K is nonempty then L K ( m ) has degree n , so ∞ A ( t ) L K ( m ) t m = � ( 1 − t ) n + 1 m = 0 for some polynomial A ( t ) of degree at most n . It is known that A ( t ) has positive coefficients. In our example ∞ ∞ 1 + t L K ( m ) t m = ( m + 1 ) 2 = � � ( 1 − t ) 3 m = 0 m = 0
If K is a convex polytope with rational points as vertices then L K ( m ) = #( mK ∩ Z n ) is a quasi-polynomial in m .
If K is a convex polytope with rational points as vertices then L K ( m ) = #( mK ∩ Z n ) is a quasi-polynomial in m . For example, if K is ( 1 2 , 1 (0 , 1 2 ) 2 ) ( 1 2 , 0) (0 , 0) � m + 1 ) 2 and � then L K ( m ) = ( 2 ∞ 1 + t 2 L K ( m ) t m = � ( 1 − t )( 1 − t 2 ) 2 . m = 0
We will also consider convex polytopes in which part of the boundary is missing. So the Ehrhart polynomial of (0 , 1) (1 , 1) (0 , 0) (1 , 0) is m ( m + 1 ) .
Richard Stanley’s theory of P-partitions connects certain Ehrhart polynomials with permutation enumeration.
Richard Stanley’s theory of P-partitions connects certain Ehrhart polynomials with permutation enumeration. We consider polytopes in the unit cube in R n cut out by hyperplanes x i = x j . More precisely, these polytopes are defined by inequalities of the form x i ≤ x j or x i > x j for i < j , together with 0 ≤ x i ≤ 1.
For example, in R 2 , we might take x 1 > x 2 and we would have the triangle (0 , 1) (1 , 1) (0 , 0) (1 , 0) The Ehrhart polynomial counts integers i 1 , i 2 satisfying � m + 1 � 0 ≤ i 2 < i 1 ≤ m and so it is equal to . 2
Sometimes a set of inequalities will be inconsistent, but when it is consistent it can be represented by a poset on [ n ] = { 1 , 2 , . . . , n } . For example, the set of inequalities x 2 < x 1 , x 2 ≤ x 3 corresponds to the poset 3 1 2
Let P be a partial order on [ n ] . A P -partition is a point ( x 1 , . . . , x n ) in R n such that 1. If i < P j then x i ≤ x j 2. If i < P j and i > j then x i < x j
Let L ( P ) be the set of extensions of P to a total order. 3 1 1 3 1 3 L ( P ) = P = 2 2 2
Let P ( P ) be the set of P -partitions. The Fundamental Theorem of P-partitions. (Stanley, Knuth, Kreweras, MacMahon) � P ( P ) = P ( π ) π ∈ L ( P )
Let P ( P ) be the set of P -partitions. The Fundamental Theorem of P-partitions. (Stanley, Knuth, Kreweras, MacMahon) � P ( P ) = P ( π ) π ∈ L ( P ) Example: 3 1 1 3 3 L ( P ) = 1 P = 2 2 2 { x 2 < x 1 ≤ x 3 } ⊔ { x 2 ≤ x 3 < x 1 } x 2 < x 1 , x 2 ≤ x 3
The order polynomial of P , denoted by Ω P ( m ) , is the Ehrhart polynomial 1 of the polytope associated to P . In other words, it is the number of P -partitions with entries in the set { 0 , 1 , . . . , m } . 1 Actually this is what Stanley calls Ω P ( m + 1 ) .
The order polynomial of P , denoted by Ω P ( m ) , is the Ehrhart polynomial 1 of the polytope associated to P . In other words, it is the number of P -partitions with entries in the set { 0 , 1 , . . . , m } . For example, if P is a disjoint union of n points, then Ω P ( m ) is ( m + 1 ) n . 1 Actually this is what Stanley calls Ω P ( m + 1 ) .
The order polynomial of P , denoted by Ω P ( m ) , is the Ehrhart polynomial 1 of the polytope associated to P . In other words, it is the number of P -partitions with entries in the set { 0 , 1 , . . . , m } . For example, if P is a disjoint union of n points, then Ω P ( m ) is ( m + 1 ) n . By the fundamental theorem of P-partitions � Ω P ( m ) = Ω π ( m ) . π ∈ L ( P ) 1 Actually this is what Stanley calls Ω P ( m + 1 ) .
The order polynomial of P , denoted by Ω P ( m ) , is the Ehrhart polynomial 1 of the polytope associated to P . In other words, it is the number of P -partitions with entries in the set { 0 , 1 , . . . , m } . For example, if P is a disjoint union of n points, then Ω P ( m ) is ( m + 1 ) n . By the fundamental theorem of P-partitions � Ω P ( m ) = Ω π ( m ) . π ∈ L ( P ) What is Ω π ( m ) ? 1 Actually this is what Stanley calls Ω P ( m + 1 ) .
First we note that linear orders on [ n ] may be identified with permutations of [ n ] : 3 1 3 1 2 2 2 1 3 2 3 1
First we note that linear orders on [ n ] may be identified with permutations of [ n ] : 3 1 3 1 2 2 2 1 3 2 3 1 � m + n � If π is the permutation 1 2 · · · n , then Ω P ( m ) = . n
First we note that linear orders on [ n ] may be identified with permutations of [ n ] : 3 1 3 1 2 2 2 1 3 2 3 1 � m + n � If π is the permutation 1 2 · · · n , then Ω P ( m ) = . n What about an arbitrary permutation?
First we note that linear orders on [ n ] may be identified with permutations of [ n ] : 3 1 3 1 2 2 2 1 3 2 3 1 � m + n � If π is the permutation 1 2 · · · n , then Ω P ( m ) = . n What about an arbitrary permutation? A descent of a permutation π of n is an i for which π ( i ) > π ( i + 1 ) . We denote by des ( π ) the number of descents of π .
� m + n − des ( π ) � Lemma. For any permutation π of n , Ω π ( m ) = . n
� m + n − des ( π ) � Lemma. For any permutation π of n , Ω π ( m ) = . n Proof by example. Consider the permutation π = 1 4 • 2 5 • 3. The π -partitions f with parts in [ m ] satisfy 0 ≤ x 1 ≤ x 4 < x 2 ≤ x 5 < x 3 ≤ m This is the same as 0 ≤ x 1 ≤ x 4 ≤ x 2 − 1 ≤ x 5 − 1 ≤ x 3 − 2 ≤ m − 2 � ( m − 2 )+ 5 � and the number of solutions of these inequalities is . 5
As a consequence we have the fundamental result for counting permutations by descents: ∞ π ∈ L ( P ) t des ( π ) � Ω P ( m ) t m = � ( 1 − t ) n + 1 m = 0 Proof. By linearity it is enough to consider the case in which P is the total order corresponding to a permutation π for which we have ∞ ∞ � m + n − des ( π ) � Ω π ( m ) t m = � � t m n m = 0 m = 0 t des ( π ) = ( 1 − t ) n + 1
So for the case of an antichain, where L ( P ) is the set S n of all permutations of [ n ] , we have ∞ E n ( t ) ( m + 1 ) n t m = � ( 1 − t ) n + 1 , m = 0 where � t des ( π ) . E n ( t ) = π ∈ S n
Signed P-partitions We now consider some more general polytopes, associated with root systems of type B (Reiner, C. Chow, Stembridge). In addi- tion to the hyperplanes x i = x j we also take the hyperplanes x i = − x j and x i = 0. More precisely we take the inequalities x i ≥ 0, x i < 0, x i + x j ≥ 0 and x i + x j ≤ 0, and we also restrict to the cube [ − 1 , 1 ] n in R n .
Signed P-partitions We now consider some more general polytopes, associated with root systems of type B (Reiner, C. Chow, Stembridge). In addi- tion to the hyperplanes x i = x j we also take the hyperplanes x i = − x j and x i = 0. More precisely we take the inequalities x i ≥ 0, x i < 0, x i + x j ≥ 0 and x i + x j ≤ 0, and we also restrict to the cube [ − 1 , 1 ] n in R n . So, for example, in R 2 we are looking at part of
Signed P-partitions We now consider some more general polytopes, associated with root systems of type B (Reiner, C. Chow, Stembridge). In addi- tion to the hyperplanes x i = x j we also take the hyperplanes x i = − x j and x i = 0. More precisely we take the inequalities x i ≥ 0, x i < 0, x i + x j ≥ 0 and x i + x j ≤ 0, and we also restrict to the cube [ − 1 , 1 ] n in R n .
We would like to represent these inequalities by posets of some kind.
We would like to represent these inequalities by posets of some kind. Let [ n ] ± = {− n , − ( n − 1 ) , . . . , − 1 , 0 , 1 , . . . , n } . A signed poset of order n is a partial order P on [ n ] ± with the property that i < P j if and only if − j < P − i . -2 -1 0 1 2
If P is a signed poset of order n , a P -partition is a ( 2 n + 1 ) -tuple ( x − n , . . . , x − 1 , x 0 , x 1 , . . . , x n ) such such that for all i ∈ [ n ] ± , x − i = − x i (which implies that x 0 = 0), together with the usual properties for a P -partition: if i < P j then x i ≤ x j and if i < P j and i > j then x i < x i .
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