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A cycle lemma for permutation inversions

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A cycle lemma for permutation inversions
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  Discrete Mathematics 257 (2002) 1–13www.elsevier.com/locate/disc A cycle lemma for permutation inversions S. Brunetti a ; ∗ , A. Del Lungo a , F. Del Ristoro  b a Dipartimento di Matematica e Informatica Teorica, Universita di Siena, Via del Capitano 15,53100 Siena, Italy  b DSI, Universita di Firenze, Via Lombroso 6  =  17, 50134, Firenze, Italy Received 20 August 1999; received in revised form 24 September 2001; accepted 8 October 2001 Abstract In this paper we study some properties of the  inversion  statistic. Some enumerative resultsconcerning the permutations of the multiset  {  x m 1 1  ;x m 2 2  }  with respect to the inversion parameter are established and it is shown that these depend on gcd( m 1 ;m 2 ). Using a “cycle lemma”, acombinatorial proof of the results is given. Moreover, some applications to the Gaussian binomialcoecient are illustrated. c   2002 Elsevier Science B.V. All rights reserved. Keywords:  Multiset; Permutation; Inversion; Major; Cyclic shift; Greatest common divisor; Congruence;Gaussian binomial coecient 1. Introduction The  inversion  statistic is a well-known statistic on permutations of sets and multisetsstudied by many authors [1,3,6,7,10,13]. A multiset is a set with possibly repeated elements. Let  X  n  = {  x m 1 1  ;x m 2 2  }  be a multiset of cardinality  n  such that  n = m 1  +  m 2 ,where  x 1 ¡x 2 . Given a word  w = w 1 w 2  :::w n  which is a permutation of   X  n , the  inversionnumber  of   w  is dened as inv( w )= |{ ( i;j ):  i¡j  and  w i ¿w  j }| . Let I n;h  = { w  ∈ X n  :inv( w ) ≡ h (mod n ) } ; with 0 6 h¡n , where  X n  is the set of all permutations of   X  n . We will illustrate that thefamily of sets  I n; 0 ; I n; 1 ;:::; I n;n − 1  is a partition of   X n , whose properties are connected ∗ Corresponding author. E-mail addresses:  sara.brunetti@unisi.it (S. Brunetti), dellungo@unisi.it (A. Del Lungo), fdr@dsi.uni.it(F. Del Ristoro).0012-365X/02/$-see front matter  c   2002 Elsevier Science B.V. All rights reserved.PII: S0012-365X(02)00254-6  2  S. Brunetti et al./Discrete Mathematics 257 (2002) 1–13 to gcd( m 1 ;m 2 ) and to Raney’s Lemma [12]. We introduce the following cycle lemma: if gcd( m 1 ;m 2 )= d  and  w = w 1 w 2  :::w n  is a permutation of   X  n , exactly  d  of thecyclic shifts:  w 1 w 2  :::w n ,  w 2  :::w n w 1 ;:::;w n w 1  :::w n − 1  have an inversion number congruent to  h  modulo  n , with  h ≡ inv( w ) +  jd (mod n ), and 0 6  j¡n=d .For any 0 6  j¡n=d , let  I n;j  =  d − 1 i =0  I n;i +  jd . By the cycle lemma, if gcd( m 1 ;m 2 )= d ,then |   I n; 0 | = |   I n; 1 | =  ···  = |   I n;n=d − 1 | =  dn   nm 1  : We wish to point out that, if gcd( m 1 ;m 2 )=1, then | I n; 0 | = | I n; 1 | =  ···  = | I n;n − 1 | = 1 n   nm 1  :  (1)A classical result of MacMahon [10] provides the following formula for the distributionof the inversion statistic over the set  X n  of all permutations of   X  n :  w ∈ X n q inv( w ) =   nm 1  q where   nm 1  q is a Gaussian binomial coecient.Therefore, by equipartition (1), the sum of the coecients of    nm 1  q =  l ¿ 0 p l ( m 1 ;m 2 ) q l with  l  congruent to  h  modulo  n  is (1 =n )  nm 1  , where 0 6 h¡n . For example, the multiset {  x m 1 1  ;x m 2 2  } , with  m 1  =7 and  m 2  =4 gives  117  q which is equal to 1+  q + 2 q 2 + 3 q 3 + 5 q 4 + 6 q 5 + 9 q 6 + 11 q 7 + 14 q 8 + 16 q 9 + 19 q 10 +20 q 11 + 23 q 12 + 23 q 13 + 24 q 14 + 23 q 15 + 23 q 16 + 20 q 17 + 19 q 18 + 16 q 19 + 14 q 20 + 11 q 21 +9 q 22 + 6 q 23 + 5 q 24 + 3 q 25 + 2 q 26 +  q 27 +  q 28 =30 30 30 30 30 30 30 30 30 30 30 and since  d =1,  n =11, we obtain:  k  ¿ 0  p h +12 k  (7 ; 4)=30=  111  117  , for each 0 6 h¡ 10.  S. Brunetti et al./Discrete Mathematics 257 (2002) 1–13  3 Then, we show a further property of the inversion statistic. Let  J m 1 ;h  be the classof permutations of   X  n  having the rst element equal to  x 1  and the inversion number congruent to  h  modulo  m 1 : J m 1 ;h  = { w =  x 1 w 2  :::w n ∈ X n  :inv( w ) ≡ h (mod m 1 ) } ; with 0 6 h¡m 1 . For any 0 6  j¡m 1 =d , let  J m 1 ;j  =  d − 1 i =0  J m 1 ;i +  jd . We have the followingsub-lemma:if gcd( m 1 ;m 2 )= d  and  w = w 1 w 2  :::w n  is a permutation of   X  n , then exactly  d  of the cyclic shifts:  w 1 w 2  :::w n ,  w 2  :::w n w 1 ;:::;w n w 1  :::w n − 1  have the rst elementequal to  x 1  and inversion number congruent to  h  modulo  m 1 , with  h ≡ inv( w ) +  jd (mod m 1 ), and 0 6  j¡m 1 =d .From this lemma, we deduce that if gcd( m 1 ;m 2 )= d , then |   J m 1 ; 0 | = |   J m 1 ; 1 | =  ···  = |   J m 1 ;m 1 =d − 1 | =  dm 1   n − 1 m 1  − 1  : Since ( d=m 1 )  n − 1 m 1  =( d=n )  nm 1  , we have the interesting result |   J m i ;h 1 | = |   I n;h 2 | =  dn   nm 1  ; where 1 6 i 6 2, 0 6 h 1 ¡m i =d  and 0 6 h 2 ¡n=d .We now give a summary of the paper. In Section 2, we prove the cycle lemma and some enumerative results. In Section 3, we show that, from the results of Section 2, the sub-lemma and a property which presents a curious analogy with the Chung–Feller Theorem follow [2,4]. We point out that this sub-lemma may be considered as a generalization of the cycle lemma dened by Narayana in [11]. Finally, in Section 4, some applications of our results concerning the Gaussian binomial coecient are found by using the result of MacMahon. 2. A cycle lemma Assume, without loss of generality, that the elements of the multiset  X  n  are  x 1  =1 and  x 2  =2 (i.e.,  X  n  = { 1 m 1 ; 2 m 2 }  and  n = m 1  + m 2 ). Let  w = w 1 w 2  :::w n  be a permutation of  { 1 m 1 ; 2 m 2 } . We dene the operator   #  in the following way:  # ( w )= w 2 w 3  :::w n w 1 , whichcarries out a cyclic shift of   w . Therefore, the inversion number of   # ( w ) is such that •  if   w 1  =1, then inv( # ( w ))=inv( w ) +  m 2 , •  if   w 1  =2, then inv( # ( w ))=inv( w ) − m 1 .Thus,  i  consecutive applications of   #  produce •  if   w i  =1, then inv( # i ( w ))=inv( # i − 1 ( w )) +  m 2 , •  if   w i  =2, then inv( # i ( w ))=inv( # i − 1 ( w )) − m 1 .  4  S. Brunetti et al./Discrete Mathematics 257 (2002) 1–13 Table 1The cyclic shifts of   w  =112111212111 and their corresponding inversion number  Cyclic shifts inv inv(mod12) w =112111212111 14 2 # 1 ( w )=121112121111 17 5 # 2 ( w )=211121211111 20 8 # 3 ( w )=111212111112 11 11 # 4 ( w )=112121111121 14 2 # 5 ( w )=121211111211 17 5 # 6 ( w )=212111112111 20 8 # 7 ( w )=121111121112 11 11 # 8 ( w )=211111211121 14 2 # 9 ( w )=111112111212 5 5 # 10 ( w )=111121112121 8 8 # 11 ( w )=111211121211 11 11 Since  − m 1 ≡ m 2  (mod n ), the previous relations giveinv( # i ( w )) ≡ inv( w ) − m 1 i (mod n ) :  (2) Now, we require to determine the number of cyclic shifts of   w  whose inversion number is congruent to  h  modulo  n , with 0 6 h 6 n  −  1, that is the number of   i  for whichinv( # i ( w )) ≡ h (mod n ), with 0 6 i¡n . This is equivalent to the determination of thenumber of solutions of congruenceinv( w ) − m 1 i ≡ h (mod n ) :  (3)Let us consider the following classical theorem [8, p. 29]: Theorem 1.  Let  gcd( a;n )= d .  Then the equation  ax ≡ b (mod n )  has no solutions if   d does not divide  b .  If   d  divides  b  then it has exactly  d  solutions .Since gcd( m 1 ;m 1  +  m 2 )=gcd( m 1 ;m 2 ), from Eq. (3) and this theorem, where  a = − m 1 ,  x = i  and  b = h − inv( w ), we deduce the following cycle lemma: Cycle Lemma 2.2.  Let  gcd( m 1 ;m 2 )= d  and   w  be a permutation of   X  n .  Exactly  d  of the cyclic shifts  w;# 1 ( w ) ;:::;# n − 1 ( w )  have inversion number congruent to  h  modulo n ,  with  h ≡ inv( w ) +  jd (mod n ),  and   0 6  j¡n=d .For example, consider the multiset  { 1 m 1 ; 2 m 2 } , with  m 1  =9 and  m 2  =3. Table 1shows the cyclic shifts of the permutation  w =112111212111 and their corres- ponding inversion number modulo  n =12. Note that inv( w )=14,  d =gcd( m 1 ;m 2 )=3 and 3 of the cyclic shifts  w;# 1 ( w ) ;:::;# 11 ( w ) have inversion number con-gruent to  h  modulo 12, with  h =2 ; 5 ; 8 ; 11 (i.e.,  h ≡ inv( w ) + 3  j (mod12) and0 6  j¡ 4).In some cases, some of the cyclic shifts of a permutation are equal. For example, the permutation  w =211121112111 is made up of a subsequence repeated three times in  w ,  S. Brunetti et al./Discrete Mathematics 257 (2002) 1–13  5Table 2The cyclic shifts of   w  =211121112111 and their corresponding inversion number  Cyclic shifts inv inv(mod12) w =211121112111 18 6 # 1 ( w )=111211121112 9 9 # 2 ( w )=112111211121 12 0 # 3 ( w )=121112111211 15 3 which we call  period   and this is  T   =2111. The number of dierent cyclic shifts of   w  isequal to the length of the period (i.e., four) and these sequences are  w =211121112111, # ( w )=111211121112,  # 2 ( w )=112111211121 and  # 3 ( w )=121112111211. In this case,inv( w )=18,  d =gcd( m 1 ;m 2 )=3 and there is only one distinct cyclic shift of   w  whoseinversion number is congruent to  h  modulo 12, with  h =6 ; 9 ; 0 ; 3 (see Table 2). We  point out that, in any case, we have the same number of distinct cyclic shifts of   w whose inversion number is congruent to  h  modulo  n , with  h ≡ inv( w ) +  jd (mod n ),and 0 6  j¡n=d .Let us denote the set  { w;# ( w ) ;:::;# ( n − 1) ( w ) }  by  S  w . Since the family  { S  w :  ∀ w ∈ X n } is a partition of the set  X n  (i.e.,  S  w  ∩ S  w ′  = ∅  or   S  w  = S  w ′  for each pair   w;w ′ ∈ X n ),from Lemma 2.2 we deduce that  #  is a bijection between  I n;i  and  I n;i +  jd , for each0 6  j¡n=d . Consequently, | I n;i | = | I n;i + d | =  ···  = | I n;i +  jd | =  ···  = | I n;i +( n=d − 1) d | for each 0 6 i¡d . The multiset  { 1 9 ; 2 3 }  of the previous example gives | I n; 0 | = | I n; 3 | = | I n; 6 | = | I n; 9 | ; | I n; 1 | = | I n; 4 | = | I n; 7 | = | I n; 10 | ; | I n; 2 | = | I n; 5 | = | I n; 8 | = | I n; 11 | : The sets  I n; 0 ;:::; I n;n − 1  are disjoint, thus if   I n;j  denote  I n;jd ∪ I n; 1+  jd ∪ ··· ∪ I n;d − 1+  jd , with 0 6  j¡n=d , we have |   I n; 0 | = |   I n; 1 | =  ···  = |   I n;n=d − 1 | : The family  I n; 0 ;   I n; 1 ;:::;   I n;n=d − 1  is a partition of the set of all the permutations of   X  n ,and so: Theorem 3.  If   gcd( m 1 ;m 2 )= d ,  then  |   I n;j | =( d=n )  nm 1  ,  where  0 6  j¡n=d . Remark 4.  The  major  of a permutation  w ,  maj ( w ), is the sum of all integers  j  suchthat 1 6  j 6 n − 1 and  w  j ¿w  j +1  (see [1]). MacMahon [10] proved that  w ∈ X n q maj ( w ) =  w ∈ X n q inv( w ) =   nm 1  q :
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