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 Π n , k = { w ∈ S n | w 1 < w 2 < · · · < w n − k , is( w ) = n − k } , where is( w ) — length of the longest increasing subsequence of the permutation w Examples is : is(315264) = 3, 3 1 5 2 6 4,3 1 5 2 6 4 . 24568 317 ∈ Π 8 , 3 Π 4 , 2 = { 1432 , 2413 , 2431 , 3412 , 3421 } Π 5 , 2 = { 12543 , 13524 , 13542 , 14523 , 14532 , 23514 , 23541 , 24513 , 24531 , 34512 , 34521 } #Π n , k =? Answer (A.Garsia, A.Goupil) k � k � n ! � ( − 1) k − r #Π n , k = r ( n − r )! r =0 for n ≥ 2 k. Proof. (A.Garsia, A.Goupil, Character Polynomials, their q-analogues and the Kronecker product ) H m = � r ( − 1) r h m + r e ⊥ Schur row adder: r , s ( n − k ,µ ) = H n − k s µ � 1 − q ] ,φ n , k = � ˜ λ ⊢ n s λ [ X ] s λ [ 1 µ ⊢ k f µ s ( n − k ,µ ) = H n − k e k H n ( x ; q )=( q , q ) n 1 H n � =Π n , k ( q )= � ( ··· = � φ n , k , ˜ k s ) ( − 1) k − s � h n − s e s 1 , ˜ H n � = ···

The Problem: Find a bijective proof of Π n , k formula! The Problem (A.Garisa, A.Goupil) Show k � k � n ! � ( − 1) k − r #Π n , k = r ( n − r )! r =0 bijectively (‘elementary’)! $ 100 award offered! (Garsia) Bijection for its q − analogue also: Theorem (A.Garsia, A.Goupil) k � k � � � q maj( w − 1 ) = ( − 1) k − r [ n ] q · · · [ n − r + 1] q , r w ∈ Π n , k r =0 where maj( σ ) = � i | σ i >σ i +1 i denotes the major index of a permutation and [ n ] q = 1 − q n 1 − q Background Partition of n : λ ⊢ n , λ = ( λ 1 ≥ λ 2 ≥ · · · ) � λ i = n Young diagram of shape λ : Standard Young Tableau of shape λ : λ = (4 , 2 , 2) → < 1 3 4 8 ∧ 2 6 5 7

Robinson-Schensted-Knuth Theorem RSK is a bijection between permutations w ∈ S n and pairs of Standard Young Tableaux of same shape with n elements: w → ( P Q ) , ���� ���� Insertion Tableau Recording Tableau . RSK algorithm, step i : w 1 . . . w i → ( P i , Q i ) P i +1 = w i +1 → P i : 5 → 1 3 7 1 3 5 1 3 5 = 7 → = 2 2 2 7 4 4 4 Q i +1 = Q i + i+1 @new box of P i +1 Example (of RSK) w 1 = 5 , 5 6 56 1 , � � 1 6 , 1 2 � � � � 5 , 1 5 6 , 1 2 5 3 w = 561423 561 4 , 5614 2 , 56142 3 ,     1 2 1 2 1 2 3 1 2 6 � � 5 6 , 1 2 1 4 , , 4 6 3 4 4 6 3 4     3 4 5 5 5 5

Some properties of RSK Theorem (Schensted) If w − → ( P , Q ) and λ = sh( P ) , then λ 1 = is( w ) .   1 2 3 1 2 6 is(561423) = 3 , 561423 − → , 4 6 3 4   5 5   1 2 3  = (3 , 2 , 1) , λ 1 = 3 λ = shape 4 6  5 Bijections Setup Lemma w ∈ Π n , k = { w ∈ S n | w 1 < w 2 < · · · < w n − k , is( w ) = n − k } ⇐ ⇒   . . . . 1 2 . . . n-k w − →  .  . . . . . . . . . . , Definition C n , k := { w ∈ S n | w 1 < w 2 < · · · < w n − k } . � n � # C n , k = k ! k Lemma   . . . . . 1 2 . . . n-k . w ∈ C n , k ⇐ ⇒ w − →  .  . . . . . . . . . . ,

An easy bijection n − k n − k 1 2 . . . n-k . 1 2 . . . n-k a 1 . . . 1 2 . . . n-k +1 . . . a s + s = − → . . . . . . . . . . . . . . . . . . Map: ( a 1 , . . . , a s ) → ( n − k + 1 , . . . , n − k + s ) and all elements below first row: [ n − k + 1 , . . . , n ] \ [ a 1 , . . . , a s ] → [ n − k + s + 1 , . . . , n ] , preserving the order. n − k n − k 1 2 . . . n-k a 1 . . . a s 1 2 . . . n-k +1 . . . + s ← → × ( a 1 , . . . , a s ) . . . . . . . . . . . . Example ( n = 9 , k = 5) 1 2 3 4 6 8 1 2 3 4 5 6 − → × ( 6 , 8 ) 5 9 7 9 7 8                 . . . . . . . 1 2 . . . n-k .       C n , k ← → , . . . . . . . .         . . . .         � �� �   P     n − k � [ n − k + 1 , . . . , n ] � 1 2 . . . n-k .  + s  � ↔  P , ×  . . . . s   s . . � �� � Π n , k − s Lemma � [ k ] � � n � � k � As sets: C n , k ≃ � k k ! = � k s =0 Π n , k − s × . As numbers: s =0 Π n , s × . s k s

Inclusion-Exclusion bijection     a 1 > · · · > a s , > < 1 2 . . . n-k a 1 . . . a s b 1 . . .     b 1 < b 2 < . . . , D n , k , s : = { P , } , . . . .   <   rest ∧ ↓ , → . .   � �� � Q   1 2 . . . n-k a 1 . . . C n , k \ Π n , k = {  P ,  } ⊂ D n , k , 1 . . . . . . � �� � E n , k , 1     1 . . . n-k a 1 a 2 .   D n , k , 1 \ E n , k , 1 =  P , ⊂ D n , k , 2 , a 1 > a 2  . . .   . . � �� � E n , k , 2 . . . Π n , k = C n , k \ ( D n , k , 1 \ ( D n , k , 2 \ . . . D n , k , k ))     n − k n − k 1 2 . . . n-k +1 . . . b 1 .  + s   × ( a 1 , . . . , a s ) D n , k , s ≃  P , . . . .   . . � [ k ] � = C n , k − s × s Theorem When 2 k ≤ n, k � k � k � k � n ! � � ( − 1) s # C n , k − s ( − 1) k − r #Π n , k = = ( n − r )! . s r s =0 r =0

The major index Definition The descent set of an SYT T is   . . i .   D ( T ) =  i : = T  . . . . . i+1 . The major index is maj( T ) = � i ∈ D ( T ) i . 1 3 4 8 T = D ( T ) = { 1 , 4 , 6 } , maj( T ) = 11 , 2 5 6 7 Corollary (to RSK) RSK ↔ ( P , Q ) , then D ( w − 1 ) = D ( P ) , so maj( w − 1 ) = maj( P ) . If w q − analogue � � q maj( w − 1 ) = q maj( P ) w ∈ Π n , k ( P , Q ) ∈ RSK (Π n , k ) � q maj( P ) = ( P , Q ) ∈ C n , k \ ( D n , k , 1 \ ( D n , k , 2 \ ... D n , k , k )) � � � q maj( P ) − q maj( P ) + q maj( P ) + . . . = ( P , Q ) ∈ C n , k ( P , Q ) ∈ D n , k , 1 ( P , Q ) ∈ D n , k , 2   1 . . . n-k a 1 . . . a s . . . ( P , Q ) =  P ,  . . . . .   n − k n − k 1 . . . n-k +1 . . . . . . + s  × ( a 1 , . . . , a s ) ↔  P , . . . . . � [ k ] � D n , k , s ≃ { ( P , Q ′ ) ∈ C n , k − s } × s

� q maj( P ) = ( P , Q ) ∈ C n , k � k � � k � � � q maj( P ) + q maj( P ) + . . . − 1 2 ( P , Q ) ∈ C n , k − 1 ( P , Q ) ∈ C n , k − 2 But ( P , Q ) ∈ D n , k , s ⇔ Q ↔ Q ′ × ( a 1 , . . . , a s ) , Q ′ ∈ C n , k − s , so � � � q maj( P ) − q maj( P ) + q maj( P ) + . . . = ( P , Q ) ∈ C n , k ( P , Q ) ∈ C n , k − 1 ( P , Q ) ∈ C n , k − 2 � q maj( P ) = ( P , Q ) ∈ C n , k � k � � k � � � q maj( P ) + q maj( P ) + . . . − 1 2 ( P , Q ) ∈ C n , k − 1 ( P , Q ) ∈ C n , k − 2 Lemma � q maj( w − 1 ) = [ n ] q . . . [ n − r + 1] q , w ∈ C n , r Proof: P − partitions or Foata’s bijection. Theorem k � k � � � q maj( w − 1 ) = ( − 1) k − r [ n ] q · · · [ n − r + 1] q , r r =0 w ∈ Π n , k

� � � � � � Permutations only, no RSK, definitions Issue: Cannot apply RSK − 1 to the non-SSYTs in D n , k , s . Hence we need a slightly different approach: Definition LLI- m (Least Lexicographic Indices) property of an increasing subsequence σ = w i 1 , w i 2 , . . . , w i m of w ∈ S n if is( w 1 . . . w i m − 1 ) = m − 1 ( i 1 , . . . , i m ) - lexicographically smallest such w = 12684357’s LLI-5 12 68 4 3 57 . Definition φ : C n , s \ Π n , s → C n , s − 1 × [ n − s + 1 , . . . , n ] : LLI- n − s + 1 sequence of w : w 1 . . . w r w i r +1 . . . w i n − s +1 , φ ( w ) = ( w 1 . . . w r w r +1 . . . w n − s . . . w i r +1 . . . w i r + w . . . ) × i m 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 , 11 2 , 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 C n , s , a = { w ∈ C n , s | LLI − ( n − s + 1) ends at index a } . Lemma φ is injective. C n , s − 1 \ φ ( C n , s , a ) = ( � n − s +2 ≤ b ≤ a C n , s − 1 , b ) × a.

Inclusion-Exclusion bijection on permutations � � C n , k \ Π n , k = C n , k , a 1 ≃ Φ( C n , k , a 1 ) n − k +1 ≤ a 1 ≤ n n − k +1 ≤ a 1 ≤ n   � � =  C n , k − 1 × a 1 \ C n , k − 1 , a 2  n − k +1 ≤ a 1 ≤ n n − k +2 ≤ a 2 ≤ a 1   � [ k ] � � = C n , k − 1 × \ C n , k − 1 , a 2 × a 1   1 n − k +2 ≤ a 2 ≤ a 1 ≤ n � [ k ] � ≃ C n , k − 1 × \ 1     � [ k ] � �  C n , k − 1 × \ C n , k − 2 , a 3 × ( a 2 , a 1 )    2 n − k +3 ≤ a 3 ≤ a 2 ≤ a 1 ≤ n � [ k ] � � � [ k ] � C n , k \ Π n , k ≃ C n , k − 1 × \ C n , k − 2 × \ . . . 1 2 � � [ k ] � � � · · · \ C n , k − r × \ . . . . . . r k � k � n ! � ( − 1) k − r = ⇒ #Π n , k = r ( n − r )! r =0 Also maj( w − 1 ) = maj( φ ( w ) − 1 ), so maj is preserved and k � k � � � q maj( w − 1 ) = ( − 1) k − r [ n ] q · · · [ n − r + 1] q r w ∈ Π n , k r =0