Bijective enumeration of permutations starting with a longest - - PDF document

Bijective enumeration of permutations starting with a longest increasing subsequence

Greta Panova

Harvard

FPSAC 2010

The Main Objects of Interest

Definition Πn,k = {w ∈ Sn|w1 < w2 < · · · < wn−k, is(w) = n − k}, where is(w) — length of the longest increasing subsequence of the permutation w Examples is : is(315264) = 3, 315264,315264. 24568317 ∈ Π8,3 Π4,2 = {1432, 2413, 2431, 3412, 3421} Π5,2 = {12543, 13524, 13542, 14523, 14532, 23514, 23541, 24513, 24531, 34512, 34521}

#Πn,k =?

Answer (A.Garsia, A.Goupil) #Πn,k =

k

• r=0

(−1)k−r k r

• n!

(n − r)! for n ≥ 2k. Proof. (A.Garsia, A.Goupil, Character Polynomials, their q-analogues and the Kronecker product)

Schur row adder: Hm=

r(−1)rhm+re⊥ r ,s(n−k,µ)=Hn−ksµ

˜ Hn(x;q)=(q,q)n

• λ⊢n sλ[X]sλ[

1 1−q],φn,k= µ⊢k fµs(n−k,µ)=Hn−kek 1

···=φn,k,˜ Hn=Πn,k(q)= (

k s)(−1)k−shn−ses 1,˜

Hn=···

The Problem: Find a bijective proof of Πn,k formula!

The Problem (A.Garisa, A.Goupil) Show #Πn,k =

k

• r=0

(−1)k−r k r

• n!

(n − r)! bijectively (‘elementary’)! $ 100 award offered! (Garsia) Bijection for its q−analogue also: Theorem (A.Garsia, A.Goupil)

• w∈Πn,k

qmaj(w −1) =

k

• r=0

(−1)k−r k r

• [n]q · · · [n − r + 1]q,

where maj(σ) =

i|σi>σi+1 i denotes the major index of a permutation and

[n]q = 1−qn

1−q

Background

Partition of n: λ ⊢ n, λ = (λ1 ≥ λ2 ≥ · · · ) λi = n Young diagram of shape λ: λ = (4, 2, 2) → Standard Young Tableau

• f

shape λ: ∧

<

1 3 4 8 2 6 5 7

Robinson-Schensted-Knuth

Theorem RSK is a bijection between permutations w ∈ Sn and pairs of Standard Young Tableaux of same shape with n elements: w → ( P

• Insertion Tableau

, Q

• Recording Tableau

) . RSK algorithm, step i: w1 . . . wi → (Pi, Qi) Pi+1 = wi+1 → Pi: 5 → 1 3 7 2 4 = 7 → 1 3 5 2 4 = 1 3 5 2 7 4 Qi+1 = Qi + i+1 @new box of Pi+1 Example (of RSK) w = 561423 w1 = 5, 56 561,

• 5 , 1
• 5 6 , 1 2
• 1 6

5 , 1 2 3

• 5614,

56142, 561423,

• 1 4

5 6 , 1 2 3 4

• 

 1 2 4 6 5 , 1 2 3 4 5     1 2 3 4 6 5 , 1 2 6 3 4 5  

Some properties of RSK

Theorem (Schensted) If w − → (P, Q) and λ = sh(P), then λ1 = is(w). is(561423) = 3, 561423 − →   1 2 3 4 6 5 , 1 2 6 3 4 5   λ = shape   1 2 3 4 6 5   = (3, 2, 1), λ1 = 3

Bijections Setup

Lemma w ∈ Πn,k = {w ∈ Sn|w1 < w2 < · · · < wn−k, is(w) = n − k} ⇐ ⇒ w − →  

. . . . . . . . . , 1 2 . . . n-k . . . . .

 . Definition Cn,k := {w ∈ Sn|w1 < w2 < · · · < wn−k}. #Cn,k = n

k

• k!

Lemma w ∈ Cn,k ⇐ ⇒ w − →  

. . . . . . . . . . , 1 2 . . . n-k . . . . . .

 .

An easy bijection

1 2 . . . n-k . . . . . . .

=

1 2 . . . n-k a1 . . . as . . . . . .

− →

1 2 . . . n-k

n−k +1 . . . n−k +s

. . . . . .

Map: (a1, . . . , as) → (n − k + 1, . . . , n − k + s) and all elements below first row: [n − k + 1, . . . , n] \ [a1, . . . , as] → [n − k + s + 1, . . . , n], preserving the order.

1 2 . . . n-k a1 . . . as . . . . . .

← →

1 2 . . . n-k

n−k +1 . . . n−k +s

. . . . . .

× (a1, . . . , as) Example (n = 9, k = 5)

1 2 3 4 6 8 5 9 7

− →

1 2 3 4 5 6 7 9 8

× (6, 8) Cn,k ← →                    

. . . . . . . . . . . . .

• P

,

1 2 . . . n-k . . . . . . .

                    ↔

• s

    P,

1 2 . . . n-k .

n−k +s

. . . . . .

    

• Πn,k−s

× [n − k + 1, . . . , n] s

• Lemma

As sets: Cn,k ≃ k

s=0 Πn,k−s ×

[k]

s

• . As numbers:

n

k

• k! = k

s=0 Πn,s ×

k

s

• .

Inclusion-Exclusion bijection

Dn,k,s : = {        P,

1 2 . . . n-k a1

>

. . . as b1

<

. . . . . . . . .

• Q

       }, a1 > · · · > as, b1 < b2 < . . . , rest ∧ ↓,

<

→ Cn,k \ Πn,k = {  P,

1 2 . . . n-k a1 . . . . . . . . .

 }

• En,k,1

⊂ Dn,k,1 Dn,k,1 \ En,k,1 =     P,

1 . . . n-k a1 a2 . . . . . .

    

• En,k,2

⊂ Dn,k,2, a1 > a2 . . . Πn,k = Cn,k \ (Dn,k,1 \ (Dn,k,2 \ . . . Dn,k,k)) Dn,k,s ≃     P,

1 2 . . . n-k

n−k +1 . . . n−k +s

b1 . . . . . . .

  × (a1, . . . , as)    = Cn,k−s × [k] s

• Theorem

When 2k ≤ n, #Πn,k =

k

• s=0

(−1)s#Cn,k−s k s

• =

k

• r=0

(−1)k−r k r

• n!

(n − r)!.

The major index

Definition The descent set of an SYT T is D(T) =   i :

. . i . . . . . i+1 .

= T    . The major index is maj(T) =

i∈D(T) i.

T = 1 3 4 8 2 5 6 7 , D(T) = {1, 4, 6}, maj(T) = 11 Corollary (to RSK) If w

RSK

↔ (P, Q), then D(w −1) = D(P), so maj(w −1) = maj(P).

q−analogue

• w∈Πn,k

qmaj(w −1) =

• (P,Q)∈RSK(Πn,k)

qmaj(P) =

• (P,Q)∈Cn,k\(Dn,k,1\(Dn,k,2\...Dn,k,k))

qmaj(P) =

• (P,Q)∈Cn,k

qmaj(P) −

• (P,Q)∈Dn,k,1

qmaj(P) +

• (P,Q)∈Dn,k,2

qmaj(P) + . . . (P, Q) =  P,

1 . . . n-k a1 . . . as . . . . . . . .

  ↔  P,

1 . . . n-k

n−k +1 . . . n−k +s

. . . . . . . .

  × (a1, . . . , as) Dn,k,s ≃ {(P, Q′) ∈ Cn,k−s} × [k] s

=

• (P,Q)∈Cn,k

qmaj(P) −

• (P,Q)∈Cn,k−1

k 1

• qmaj(P) +
• (P,Q)∈Cn,k−2

k 2

• qmaj(P) + . . .

But (P, Q) ∈ Dn,k,s ⇔ Q ↔ Q′ × (a1, . . . , as), Q′ ∈ Cn,k−s, so =

• (P,Q)∈Cn,k

qmaj(P) −

• (P,Q)∈Cn,k−1

qmaj(P) +

• (P,Q)∈Cn,k−2

qmaj(P) + . . . =

• (P,Q)∈Cn,k

qmaj(P) −

• (P,Q)∈Cn,k−1

k 1

• qmaj(P) +
• (P,Q)∈Cn,k−2

k 2

• qmaj(P) + . . .

Lemma

• w∈Cn,r

qmaj(w −1) = [n]q . . . [n − r + 1]q, Proof: P−partitions or Foata’s bijection. Theorem

• w∈Πn,k

qmaj(w −1) =

k

• r=0

(−1)k−r k r

• [n]q · · · [n − r + 1]q,

Permutations only, no RSK, definitions

Issue: Cannot apply RSK −1 to the non-SSYTs in Dn,k,s. Hence we need a slightly different approach: Definition LLI-m (Least Lexicographic Indices) property of an increasing subsequence σ = wi1, wi2, . . . , wim of w ∈ Sn if is(w1 . . . wim−1) = m − 1 (i1, . . . , im) - lexicographically smallest such w = 12684357’s LLI-5 12684357. Definition φ : Cn,s \ Πn,s → Cn,s−1 × [n − s + 1, . . . , n] : LLI-n − s + 1 sequence of w: w1 . . . wrwir+1 . . . win−s+1, φ(w) = (w1. . .wrwr+1. . .wn−s. . .wir+1. . .wir+w . . .

• ) × im

n = 13, s = 7 φ(2, 4, 5, 6, 9, 13, 8, 1, 7, 10, 3, 12, 11) : 2,4, 5,6, 9,13,8,1, 7,10,3,12,112,4, 5,6, 9,13,8,1, 7,10,3,12,11

• = (2, 4, 5, 6, 8, 9, 13, 10, 1, 7, 12, 3, 11) × 12

Set Cn,s,a = {w ∈ Cn,s|LLI − (n − s + 1) ends at index a}. Lemma φ is injective. Cn,s−1 \ φ(Cn,s,a) = (

n−s+2≤b≤a Cn,s−1,b) × a.

Inclusion-Exclusion bijection on permutations

Cn,k \ Πn,k =

• n−k+1≤a1≤n

Cn,k,a1 ≃

• n−k+1≤a1≤n

Φ(Cn,k,a1) =

• n−k+1≤a1≤n

 Cn,k−1 × a1 \

• n−k+2≤a2≤a1

Cn,k−1,a2   = Cn,k−1 × [k] 1

• \

 

• n−k+2≤a2≤a1≤n

Cn,k−1,a2 × a1   ≃ Cn,k−1 × [k] 1

• \

 Cn,k−1 × [k] 2

• \

 

• n−k+3≤a3≤a2≤a1≤n

Cn,k−2,a3 × (a2, a1)     Cn,k \ Πn,k ≃ Cn,k−1 × [k] 1

• \
• Cn,k−2 ×

[k] 2

• \ . . .

· · · \

• Cn,k−r ×

[k] r

• \ . . .
• . . .
• =

⇒ #Πn,k =

k

• r=0

(−1)k−r k r

• n!

(n − r)! Also maj(w −1) = maj(φ(w)−1), so maj is preserved and

• w∈Πn,k

qmaj(w −1) =

k

• r=0

(−1)k−r k r

• [n]q · · · [n − r + 1]q