Flag Descents and P-Partitions Ira M. Gessel Department of - - PowerPoint PPT Presentation

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 Rn with lattice points as vertices. Then for any positive integer m, #(mK ∩ Zn) is a polynomial LK(m) in m, called the Ehrhart polynomial of K.

Ehrhart polynomials

Let K be convex polytope in Rn with lattice points as vertices. Then for any positive integer m, #(mK ∩ Zn) is a polynomial LK(m) in m, called the Ehrhart polynomial of K. For example, if K is the unit square in R2

(0, 0) (1, 0) (0, 1) (1, 1)

then LK(m) = (m + 1)2.

If the interior of K is nonempty then LK(m) has degree n, so

∞

• m=0

LK(m)tm = A(t) (1 − t)n+1 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 LK(m) has degree n, so

∞

• m=0

LK(m)tm = A(t) (1 − t)n+1 for some polynomial A(t) of degree at most n. It is known that A(t) has positive coefficients. In our example

∞

• m=0

LK(m)tm =

∞

• m=0

(m + 1)2 = 1 + t (1 − t)3

If K is a convex polytope with rational points as vertices then LK(m) = #(mK ∩ Zn) is a quasi-polynomial in m.

If K is a convex polytope with rational points as vertices then LK(m) = #(mK ∩ Zn) is a quasi-polynomial in m. For example, if K is

(0, 0) (0, 1

2)

( 1

2, 1 2)

( 1

2, 0)

then LK(m) = ( m

2

• + 1)2 and

∞

• m=0

LK(m)tm = 1 + t2 (1 − t)(1 − t2)2 .

We will also consider convex polytopes in which part of the boundary is missing. So the Ehrhart polynomial of

(0, 0) (1, 0) (0, 1) (1, 1)

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 Rn cut out by hyperplanes xi = xj. More precisely, these polytopes are defined by inequalities of the form xi ≤ xj or xi > xj for i < j, together with 0 ≤ xi ≤ 1.

For example, in R2, we might take x1 > x2 and we would have the triangle

(0, 0) (1, 0) (0, 1) (1, 1)

The Ehrhart polynomial counts integers i1, i2 satisfying 0 ≤ i2 < i1 ≤ m and so it is equal to m+1

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 x2 < x1, x2 ≤ x3 corresponds to the poset

1 2 3

Let P be a partial order on [n]. A P-partition is a point (x1, . . . , xn) in Rn such that

• 1. If i <P j then xi ≤ xj
• 2. If i <P j and i > j then xi < xj

Let L (P) be the set of extensions of P to a total order.

1 2 3 L (P) = 2 2 1 1 3 3 P =

Let P(P) be the set of P-partitions. The Fundamental Theorem of P-partitions. (Stanley, Knuth, Kreweras, MacMahon) P(P) =

• π∈L (P)

P(π)

Let P(P) be the set of P-partitions. The Fundamental Theorem of P-partitions. (Stanley, Knuth, Kreweras, MacMahon) P(P) =

• π∈L (P)

P(π) Example:

1 2 3 L (P) = 2 2 1 1 3 3 P = x2 < x1, x2 ≤ x3 {x2 < x1 ≤ x3} ⊔ {x2 ≤ x3 < x1}

The order polynomial of P, denoted by ΩP(m), is the Ehrhart polynomial1 of the polytope associated to P. In other words, it is the number of P-partitions with entries in the set {0, 1, . . . , m}.

1Actually this is what Stanley calls ΩP(m + 1).

The order polynomial of P, denoted by ΩP(m), is the Ehrhart polynomial1 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 ΩP(m) is (m + 1)n.

1Actually this is what Stanley calls ΩP(m + 1).

The order polynomial of P, denoted by ΩP(m), is the Ehrhart polynomial1 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 ΩP(m) is (m + 1)n. By the fundamental theorem of P-partitions ΩP(m) =

• π∈L (P)

Ωπ(m).

1Actually this is what Stanley calls ΩP(m + 1).

The order polynomial of P, denoted by ΩP(m), is the Ehrhart polynomial1 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 ΩP(m) is (m + 1)n. By the fundamental theorem of P-partitions ΩP(m) =

• π∈L (P)

Ωπ(m). What is Ωπ(m)?

1Actually this is what Stanley calls ΩP(m + 1).

First we note that linear orders on [n] may be identified with permutations of [n]:

2 2 1 1 3 3 2 1 3 2 3 1

First we note that linear orders on [n] may be identified with permutations of [n]:

2 2 1 1 3 3 2 1 3 2 3 1

If π is the permutation 1 2 · · · n, then ΩP(m) = m+n

n

• .

First we note that linear orders on [n] may be identified with permutations of [n]:

2 2 1 1 3 3 2 1 3 2 3 1

If π is the permutation 1 2 · · · n, then ΩP(m) = m+n

n

• .

What about an arbitrary permutation?

First we note that linear orders on [n] may be identified with permutations of [n]:

2 2 1 1 3 3 2 1 3 2 3 1

If π is the permutation 1 2 · · · n, then ΩP(m) = m+n

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

• f π.

• Lemma. For any permutation π of n, Ωπ(m) =

m+n−des(π)

n

• .

• Lemma. For any permutation π of n, Ωπ(m) =

m+n−des(π)

n

• .

Proof by example. Consider the permutation π = 1 4 •2 5 •3. The π-partitions f with parts in [m] satisfy 0 ≤ x1 ≤ x4 < x2 ≤ x5 < x3 ≤ m This is the same as 0 ≤ x1 ≤ x4 ≤ x2 − 1 ≤ x5 − 1 ≤ x3 − 2 ≤ m − 2 and the number of solutions of these inequalities is (m−2)+5

5

• .

As a consequence we have the fundamental result for counting permutations by descents:

∞

• m=0

ΩP(m)tm =

• π∈L (P) tdes(π)

(1 − t)n+1

• 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=0

Ωπ(m)tm =

∞

• m=0

m + n − des(π) n

• tm

= tdes(π) (1 − t)n+1

So for the case of an antichain, where L (P) is the set Sn of all permutations of [n], we have

∞

• m=0

(m + 1)ntm = En(t) (1 − t)n+1 , where En(t) =

• π∈Sn

tdes(π).

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 xi = xj we also take the hyperplanes xi = −xj and xi = 0. More precisely we take the inequalities xi ≥ 0, xi < 0, xi + xj ≥ 0 and xi + xj ≤ 0, and we also restrict to the cube [−1, 1]n in Rn.

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 xi = xj we also take the hyperplanes xi = −xj and xi = 0. More precisely we take the inequalities xi ≥ 0, xi < 0, xi + xj ≥ 0 and xi + xj ≤ 0, and we also restrict to the cube [−1, 1]n in Rn. So, for example, in R2 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 xi = xj we also take the hyperplanes xi = −xj and xi = 0. More precisely we take the inequalities xi ≥ 0, xi < 0, xi + xj ≥ 0 and xi + xj ≤ 0, and we also restrict to the cube [−1, 1]n in Rn.

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

• f 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

1

• 2

If P is a signed poset of order n, a P-partition is a (2n + 1)-tuple (x−n, . . . , x−1, x0, x1, . . . , xn) such such that for all i ∈ [n]±, x−i = −xi (which implies that x0 = 0), together with the usual properties for a P-partition: if i <P j then xi ≤ xj and if i <P j and i > j then xi < xi.

If P is a signed poset of order n, a P-partition is a (2n + 1)-tuple (x−n, . . . , x−1, x0, x1, . . . , xn) such such that for all i ∈ [n]±, x−i = −xi (which implies that x0 = 0), together with the usual properties for a P-partition: if i <P j then xi ≤ xj and if i <P j and i > j then xi < xi. Since (x−n, . . . , xn) is determined by (x1, . . . xn), we can think of a P-partition as a point in either R2n+1 or Rn.

For the poset

2

• 1

1

• 2

a P-partition is a 5-tuple (x−2, x−1, x0, x1, x2) satisfying x−i = −xi for all i, 0 < x−2, and x−1 < x−2.

The set L (P) of signed linear extensions of P is the set of linear extensions of P that are signed posets.

2

• 1

1

• 2

P = L (P)

• 2
• 2
• 2
• 1
• 1
• 1

1 1 1 2 2 2

Just as for ordinary P-partitions, we have P(P) =

• π∈L (P)

P(π) and we can identify the linear extensions of P with signed permutations.

• 2
• 1

1 2

• 1 2 0 -2 1
• r
• 2 1

We define the order polynomial ΩP(m) for a signed poset P to be the Ehrhart polynomial of the corresponding polytope, which is the number of P-partitions with values in the set [m]±. Then as before we have

∞

• m=0

ΩP(m)tm =

• π∈L (P) tdes(π)

(1 − t)n+1 , where a descent of a signed permutation π of [n] is an i ∈ {0, 1, . . . , n − 1} such that π(i) > π(i + 1). For example

• ¯

4 ¯ 1 2 • ¯ 5 3 writing ¯ i for −i.

As an example, if P is an antichain, then L (P) is the set Bn of all signed permutations of [n] and thus

∞

• m=0

(2m + 1)ntm =

• π∈Bn tdes(π)

(1 − t)n+1 . (Steingrímsson)

Flag Descents

In 2001, Adin, Brenti, and Roichman introduced a variation of the descent number of a signed permutation. They defined the flag descent number fdes(π) by fdes(π) =

• 2 des(π),

if π(1) > 0 2 des(π) − 1, if π(1) < 0 In other words, a descent at the beginning of a signed permutation contributes 1 to the flag descent number, but a descent anywhere else contributes 2.

Adin, Brenti, and Roichman showed that

∞

• m=0

(m + 1)ntm =

• π∈Bn tfdes(π)

(1 − t)(1 − t2)n , which implies that

π∈Bn tfdes(π) = (1 + t)nEn(t).

(Recently proved nicely by Lai and Petersen.)

Adin, Brenti, and Roichman showed that

∞

• m=0

(m + 1)ntm =

• π∈Bn tfdes(π)

(1 − t)(1 − t2)n , which implies that

π∈Bn tfdes(π) = (1 + t)nEn(t).

(Recently proved nicely by Lai and Petersen.) We will see that flag descents arise from Ehrhart polynomials.

Given a signed poset P, let KP be the polytope in [−1, 1]n associated to P, and let K flag

P

be obtained by shifting KP to [0, 1]n, i.e. K flag

P

= { 1

2(x1 + 1), . . . , 1 2(xn + 1)

• : (x1, . . . , xn) ∈ KP}

and we define the flag order quasi-polynomial Ωflag

P (m) to be

the Ehrhart quasi-polynomial of K flag

P

.

Given a signed poset P, let KP be the polytope in [−1, 1]n associated to P, and let K flag

P

be obtained by shifting KP to [0, 1]n, i.e. K flag

P

= { 1

2(x1 + 1), . . . , 1 2(xn + 1)

• : (x1, . . . , xn) ∈ KP}

and we define the flag order quasi-polynomial Ωflag

P (m) to be

the Ehrhart quasi-polynomial of K flag

P

.

◮ If P is an antichain, then Ωflag P (m) = (m + 1)n, ◮ In general Ωflag P (m) is the number of P-partitions with

values from −m to m, all with the same parity as m.

What is the flag order quasi-polynomial for a total order, i.e., for a signed permutation?

What is the flag order quasi-polynomial for a total order, i.e., for a signed permutation?

• Lemma. For any signed permutation π in Bn,

∞

• m=0

Ωflag

π (m)tm =

tfdes(π) (1 − t)(1 − t2)n .

Proof by example. Let’s consider the permutation π = ¯ 1¯ 342 with flag descent number 5 ( •¯ 1 •¯ 34 •2). We want to count 5-tuples (x¯

1, x¯ 3, x4, x2, m) satisfying

0 < x¯

1 < x¯ 3 ≤ x4 < x2 ≤ m,

where all entries have the same parity. Each such 5-tuple has weight tm. There is a minimal 5-tuple, (1, 3, 3, 5, 5) in which m = fdes(π). We get all such 5-tuples by adding any multiples

• f (1, 1, 1, 1, 1), (0, 2, 2, 2, 2), (0, 0, 2, 2, 2), (0, 0, 0, 2, 2), or

(0, 0, 0, 0, 2). So there’s one way to increase m by 1 and four ways to increase m by 2.

Proof by example. Let’s consider the permutation π = ¯ 1¯ 342 with flag descent number 5 ( •¯ 1 •¯ 34 •2). We want to count 5-tuples (x¯

1, x¯ 3, x4, x2, m) satisfying

0 < x¯

1 < x¯ 3 ≤ x4 < x2 ≤ m,

where all entries have the same parity. Each such 5-tuple has weight tm. There is a minimal 5-tuple, (1, 3, 3, 5, 5) in which m = fdes(π). We get all such 5-tuples by adding any multiples

• f (1, 1, 1, 1, 1), (0, 2, 2, 2, 2), (0, 0, 2, 2, 2), (0, 0, 0, 2, 2), or

(0, 0, 0, 0, 2). So there’s one way to increase m by 1 and four ways to increase m by 2. Therefore

∞

• m=0

Ωflag

π (m)tm =

tfdes(π) (1 − t)(1 − t2)n .

• Theorem. For any signed poset P on [n]±,

∞

• m=0

Ωflag

P (m)tm =

• π∈L (P) tfdes(π)

(1 − t)(1 − t2)n . The result of Adin, Brenti, and Roichman is the special case in which P is an antichain.