Counting Permutations with Even Valleys and Odd Peaks

Transcription

1 Counting Permutations with Even Valleys and Odd Peaks Ira M. Gessel Department of Mathematics Brandeis University IMA Workshop Geometric and Enumerative Combinatorics University of Minnesota, Twin Cities November 13, 2014

2 Based on the paper Counting permutations by alternating descents, by Ira M. Gessel and Yan Zhuang, Electronic Journal of Combinatorics 21 (4), 2014, Paper #P4.23

3 Introduction Let = 1 2 n be a permutation of [n] ={1, 2,...,n}. Then i is a peak if i 1 < i > i+1 j is a valley of if j 1 > j < j+1.

4 Introduction Let = 1 2 n be a permutation of [n] ={1, 2,...,n}. Then i is a peak if i 1 < i > i+1 j is a valley of if j 1 > j < j+1. Example (peaks in blue, valleys in red):

5 On Math Overflow Liviu Nicolaescu, motivated by discrete Morse functions, asked the following question: How many permutations of [n] are there are in which every valley is even and every peak is odd?

6 On Math Overflow Liviu Nicolaescu, motivated by discrete Morse functions, asked the following question: How many permutations of [n] are there are in which every valley is even and every peak is odd? Let f (n) be the number of such permutations of [n]. We can compute the first values by brute force:

7 On Math Overflow Liviu Nicolaescu, motivated by discrete Morse functions, asked the following question: How many permutations of [n] are there are in which every valley is even and every peak is odd? Let f (n) be the number of such permutations of [n]. We can compute the first values by brute force: n f(n) For example, with n = 3 the 4 permutations are (all except 132 and 231) 123, 213, 312, 321

8 Sometimes exponential generating functions for certain classes of permutations defined by restrictions on descents have nice reciprocals. So let s look at the reciprocal of P 1 n=0 f (n)x n /n!. We find that 1 X n=0 f (n)x n /n! 1 = 1 x + 2 x 3 3! 5 x 4 6 4! + 61x 6! 272 x 7 9 7! x 9!

9 Sometimes exponential generating functions for certain classes of permutations defined by restrictions on descents have nice reciprocals. So let s look at the reciprocal of P 1 n=0 f (n)x n /n!. We find that 1 X n=0 f (n)x n /n! 1 = 1 x + 2 x 3 3! 5 x 4 6 4! + 61x 6! 272 x 7 9 7! x 9! The coefficients are Euler numbers: 1X x n E n = sec x + tan x n! n=0 = 1 + x + x ! + 2x 3! + 5x 4! + 16x 5! + 61x 6! x 7 8 7! x 8! +

10 So it seems that 1X n=0 f (n) x n n! = 1 E 1 x + E 3 x 3 3! x 4 E 4 4! + E x 6 6 6! x 7 1 E 7 7! +.

11 So it seems that 1X n=0 f (n) x n n! = 1 E 1 x + E 3 x 3 3! x 4 E 4 4! + E x 6 6 6! x 7 1 E 7 7! +. This is reminiscent of the generating function 1 x + x 3 3! x 4 4! + x 6 6! x 7 1 7! + for permutations with no increasing runs of length 3 or more, due to David and Barton (1962).

12 So it seems that 1X n=0 f (n) x n n! = 1 E 1 x + E 3 x 3 3! x 4 E 4 4! + E x 6 6 6! x 7 1 E 7 7! +. This is reminiscent of the generating function 1 x + x 3 3! x 4 4! + x 6 6! x 7 1 7! + for permutations with no increasing runs of length 3 or more, due to David and Barton (1962). Is this just a coincidence?

13 Another form of the generating function is 1 E 1 x + E 3 x 3 3! x 4 E 4 4! + E x 6 6 6! x 7 1 E 7 7! + 3 sin 1 2 x + 3 cosh 1 p 2 3x = 3 cos 1 2 x p 3 sinh 1 p. 2 3x

14 Descents of permutations Let = 1 n be a permutation of [n]. Adescent of is an i such that i > i+1. The descents of a permutation split it into increasing runs, which are maximal consecutive increasing subsequences.

15 Descents of permutations Let = 1 n be a permutation of [n]. Adescent of is an i such that i > i+1. The descents of a permutation split it into increasing runs, which are maximal consecutive increasing subsequences. For example the permutation has descent set {3, 5, 6}: The increasing runs have lengths 3, 2, 1, 2.

16 Descents of permutations Let = 1 n be a permutation of [n]. Adescent of is an i such that i > i+1. The descents of a permutation split it into increasing runs, which are maximal consecutive increasing subsequences. For example the permutation has descent set {3, 5, 6}: The increasing runs have lengths 3, 2, 1, 2. We call the sequence of increasing run lengths the descent composition of. (It is a composition of n.) The descent composition has the same information as the descent set.

17 Descents of permutations Let = 1 n be a permutation of [n]. Adescent of is an i such that i > i+1. The descents of a permutation split it into increasing runs, which are maximal consecutive increasing subsequences. For example the permutation has descent set {3, 5, 6}: The increasing runs have lengths 3, 2, 1, 2. We call the sequence of increasing run lengths the descent composition of. (It is a composition of n.) The descent composition has the same information as the descent set. We ll write D(L) for the descent set corresponding to L.

18 Let L =(L 1,...,L k ) be a composition of n. Then we write n L for the multinomial coefficient n! L 1! L k! Lemma. The number of permutations of n with descent set contained in D(L) is n L.

19 Let L =(L 1,...,L k ) be a composition of n. Then we write n L for the multinomial coefficient n! L 1! L k! Lemma. The number of permutations of n with descent set contained in D(L) is n L. Proof by example. Take L =(3, 1, 2), so n = 6 and D(L) ={3, 4}. We take 1,2,..., 6 and put them into 3 boxes with three in the first, one in the second, and two in the third:

20 Let L =(L 1,...,L k ) be a composition of n. Then we write n L for the multinomial coefficient n! L 1! L k! Lemma. The number of permutations of n with descent set contained in D(L) is n L. Proof by example. Take L =(3, 1, 2), so n = 6 and D(L) ={3, 4}. We take 1,2,..., 6 and put them into 3 boxes with three in the first, one in the second, and two in the third: We arrange the numbers in each box in increasing order and then remove the bars to get which has descent set contained in {3, 4}.

21 We can find the number of permutations of n with a given descent set by inclusion-exclusion. Let us partial order compositions of n by reverse refinement, corresponding to ordering descent sets by inclusion. So the compositions of 3 are ordered as (1,1,1) (1, 2) (2,1) (3)

22 Let (L) be the number of permutations of [n] with descent composition L, where L is a composition of n.

23 Let (L) be the number of permutations of [n] with descent composition L, where L is a composition of n. Then n L = P K applel (K ), so by inclusion-exclusion (L) = X n ( 1) l(l) l(k ), K K applel where l(l) is the number of parts of L.

24 Let (L) be the number of permutations of [n] with descent composition L, where L is a composition of n. Then n L = P K applel (K ), so by inclusion-exclusion (L) = X n ( 1) l(l) l(k ), K K applel where l(l) is the number of parts of L. We can count all sorts of sets of permutations defined by descent sets by added up (L) for appropriate L. We get exponential generating functions since n L is the coefficient of x n /n! in x L L1 Lk x := x L! L 1! L k!.

25 Alternating descents Following Denis Chebikin (2008), we define an alternating descent of a permutation to be an odd descent or an even ascent. For example the alternating descents of are 3 and 6:

26 Alternating descents Following Denis Chebikin (2008), we define an alternating descent of a permutation to be an odd descent or an even ascent. For example the alternating descents of are 3 and 6: The alternating descents of a permutation split it up into alternating runs. Each alternating run is either an up-down" permutation if it starts in an odd position or a down-up" permutation if it starts in an even position.

27 Alternating descents Following Denis Chebikin (2008), we define an alternating descent of a permutation to be an odd descent or an even ascent. For example the alternating descents of are 3 and 6: The alternating descents of a permutation split it up into alternating runs. Each alternating run is either an up-down" permutation if it starts in an odd position or a down-up" permutation if it starts in an even position. Nicolaescu s question is closely connected to alternating runs of permutations, because a permutation has all valleys even and all peaks odd if and only if all of its alternating runs have length less than 3.

28 This is because an odd valley or an even peak corresponds to an alternating run of length at least 3:

29 As Chebikin observed, the number of permutations of [n] with alternating descent set contained in D(L) is n n! := L L 1! L k! E L 1 E Lk E We prove this by putting the numbers 1,..., n into boxes with L i in the ith box, and then arranging the numbers in each box into either an up-down or down-up permutation, depending on the parity of the starting position.

30 Now let ˆ(L) be the number of permutations of [n] with alternating descent composition L, where L is a composition of n n. Then just as before, L E = P ˆ(K K applel ), so by inclusion-exclusion ˆ(L) = X n ( 1) l(l) l(k ). K K applel E

31 Now let ˆ(L) be the number of permutations of [n] with alternating descent composition L, where L is a composition of n n. Then just as before, L E = P ˆ(K K applel ), so by inclusion-exclusion ˆ(L) = X n ( 1) l(l) l(k ). K K applel E Note that n L E is the coefficient of x n /n! in E L1 x L 1 L 1! E L k x Lk L k!.

32 Therefore, if we can find an exponential generating function for some class of permutations determined by descent sets, obtained by adding up (L)x L /L! for appropriate L, then by changing x n /n! to E n x n /n! we should get the corresponding result for alternating descent sets. This will explain the connection between 1 x + x 3 3! x 4 4! + x 6 6! x 7 1 7! + and 1 E 1 x + E 3 x 3 3! x 4 E 4 4! + E x 6 6 6! x 7 1 E 7 7! +.

33 Unfortunately, just changing to x n /n! with E n x n /n! doesn t really work:

34 Unfortunately, just changing to x n /n! with E n x n /n! doesn t really work: we would like a linear map that takes to x L 1 Lk x L 1! L k!. E L1 x L 1 L 1! E L k x Lk L k!. but we can t do this since these products aren t linearly independent.

35 For example, but x 2 2! x 3 5 3! = 10x 5!, x 2 E 2 2! E x 3 3 3! 6= 10E x 5 5 5!

36 To make this work, and to compute the generating functions more efficiently, we use noncommutatative symmetric functions, which were studied extensively by Gelfand, Krob, Lascoux, Leclerc, Retakh, and Thibon in 1995 and further studied by Thibon and others since then. But we will need only some very simple properties of the noncommutative symmetric functions most of which were known earlier.

37 We work with formal power series in noncommuting variables X 1, X 2,... We define the complete noncommutative symmetric functions h n = X X i1 X i2 X in. i 1 apple applei n For any composition L =(L 1,...,L k ) of n we define which can be written as h L = h L1 h Lk h L = X L X i1 X i2 X in where the sum is over all (i 1,...,i n ) satisfying i 1 apple apple i L1, i L1+1 apple apple i L1+L {z } 2,, i L1+ +L {z } k 1+1 apple apple i n {z } L 1 L 2 L k.

38 We define the ribbon noncommutative symmetric functions r L = X L X i1 X i2 X in where the sum is over all (i 1,...,i n ) satisfying i 1 apple apple i L1 > i L1+1 apple apple i L1+L {z } 2 > > i L1+ +L {z } k 1+1 apple apple i n {z } L 1 L 2 L k. Then h L = X K applel r K, so by inclusion-exclusion, r L = X K applel( 1) l(l) l(k ) h K

39 The algebra of noncommutative symmetric functions is generated by the h L.

40 The algebra of noncommutative symmetric functions is generated by the h L. When we make the X i commute, we get ordinary symmetric functions: h L becomes the ordinary complete symmetric function h L, and r L becomes the ribbon (skew-hook, zigzag) skew Schur function.

41 We can define a homomorphism from noncommutative symmetric functions to exponential generating functions by so (h n )= x n n! (h L )= x L 1 Lk x L 1! L k! = n x n L n!. Note that for a noncommutative symmetric function f, (f ) is the coefficient of x 1 x 2 x n in the result of replacing X 1, X 2,... with commuting variables x 1, x 2,... Then by inclusion-exclusion, (r L )= (L) x n n!.

42 Similarly, we can define another homomorphism from noncommutative symmetric functions to exponential generating functions by ˆ(h n )=E n x n n! so by the inclusion-exclusion formula for ˆ we have ˆ(rL )= ˆ(L) x n n!.

43 This means that if we can find the noncommutative symmetric function generating function for some class of words determined by their descent sets (i.e., we can express it in terms of the r L ), then by applying and ˆ we can get the corresponding exponential generating functions for permutations with the corresponding descent sets or alternating descent sets.

44 This means that if we can find the noncommutative symmetric function generating function for some class of words determined by their descent sets (i.e., we can express it in terms of the r L ), then by applying and ˆ we can get the corresponding exponential generating functions for permutations with the corresponding descent sets or alternating descent sets. There are many formulas to which we can apply and ˆ, but one that gives us what we need for Nicolaescu s problem is the following:

45 Theorem 1. (G, Jackson and Aleliunas; 1977) Let w 1, w 2,... be arbitrary commuting weights. Then X X 1 1 w L r L = a n h n L n=0 where the sum on the left is over all compositions L =(L 1,...,L k ), and w L = w L1 w Lk. Here a 0, a 1,... are defined by where w 0 = 1. 1X X 1 1 a n z n = w n z n, n=0 n=0

46 If we take w i = 1 for i < m and w i = 0 for i m in Theorem 1 then we get X X 1 1 r L = (h mn h mn+1 ), L n=0 where the sum on the left is over compositions L with all parts less than m.

47 If we take w i = 1 for i < m and w i = 0 for i m in Theorem 1 then we get X X 1 1 r L = (h mn h mn+1 ), L n=0 where the sum on the left is over compositions L with all parts less than m. Applying gives David and Barton s result that the exponential generating function for permutations in which every increasing run has length less than m is 1 x + x m m! x m+1 (m + 1)! + x 2m (2m)! x 2m+1 1 (2m + 1)! +,

48 Applying ˆ gives " X 1 x E mn mn n=0 (mn)! E mn+1 x mn+1 (mn + 1)! # 1 as the exponential generating function for permutations in which every alternating run has length less than m; the case m = 3 is the solution to Nicolaescu s problem.

49 Two more special cases of Theorem 1 If we set w m = 1 and w n = 0 for n 6= m then we find that 1X X 1 1 r (m n ) = ( 1) n h mn, n=0 n=0 Applying gives Carlitz s 1973 result that X 1 ( 1) n x mn (mn)! n=0 1 is the exponential generating function for permutations in which every increasing run has length m.

50 Applying ˆ shows that X 1 ( 1) n x mn E mn (mn)! n=0 1 counts permutations in which every alternating run has length m.

51 Applying ˆ shows that X 1 ( 1) n x mn E mn (mn)! n=0 1 counts permutations in which every alternating run has length m. For m = 4, these are permutations of [4n] with descent set {2, 6, 10,...,4n 2}.

52 If we set w n = t for all n 1 in Theorem 1, then we are counting words by the number of increasing runs (which is one more than the number of descents), and we get X L apple t l(l) r L =(1 t) 1 t 1X (1 t) n h n 1, n=0 where the first sum is over all compositions L.

53 If we set w n = t for all n 1 in Theorem 1, then we are counting words by the number of increasing runs (which is one more than the number of descents), and we get X L apple t l(l) r L =(1 t) 1 t 1X (1 t) n h n 1, n=0 where the first sum is over all compositions L. Applying gives the well-known generating function for the Eulerian polynomials, where 1 + 1X A n (t) x n n! = 1 t, 1 te (1 t)x n=1 A n (t) = X 2S n t des( )+1, and des( ) is the number of descents of.

54 Applying ˆ gives where 1 + 1X n=1 Â n (t) x n n! = 1 t 1 t sec(1 t)x + tan(1 t)x, Â n (t) = X 2S n t altdes( )+1 and altdes( ) is the number of alternating descents of, a result due to Chebikin.

55 The major index and alternating major index We define the major index maj( ) of a permutation to be the sum of the descents of, and we define the alternating major index altmaj( ) to be the sum of the alternating descents of. Then there is an analogue of Theorem 1 that allows us to count permutations by major index and alternating major index. I ll just state the results:

56 1X x n P 2S n t des( ) q maj( ) 1 n! (1 t)(1 tq) (1 tq n ) = X n=0 k=0 t k ky j=0 e xqj (MacMahon/Riordan 1954/Carlitz 1975) 1X n=0 x n n! P 2S n t altdes( ) q altmaj( ) (1 t)(1 tq) (1 tq n ) = 1X k=0 t k ky j=0 sec(xq j )+tan(xq j ) (Remmel)

57 The remaining material was not presented during my talk.

58 More homomorphisms There are other homomorphisms from the ring of noncommutative symmetric functions that enable us to count permutations by other parameters.

59 More homomorphisms There are other homomorphisms from the ring of noncommutative symmetric functions that enable us to count permutations by other parameters. The simplest one allows us to count permutations by inversions. An inversion of a permutation is a pair (i, j) with i < j such that i > j. We define a homomorphism 1 on noncommutative symmetric function by 1(h n )= x n (q) n, where (q) n =(1 q)(1 q 2 ) (1 q n ). Then if L is a composition of n, we have 1(r L )= X C( )=L q inv( ) x n (q) n.

60 We define the inverse major index imaj( ) to be maj( 1 ). Then it s also true that 1(r L )= X C( )=L q imaj( ) x n (q) n.

61 We define the inverse major index imaj( ) to be maj( 1 ). Then it s also true that 1(r L )= X C( )=L q imaj( ) x n (q) n. Thus, for example, 1 x + x 3 x 4 + x 6 x (q) 3 (q) 4 (q) 6 (q) 7 is the Eulerian generating function for permutations with no increasing runs of length 3 by inversions or by inverse major index.

62 More generally, if we make the variables X i commute, we get the quasi-symmetric generating function for counting permutations by their inverse descent set, from which we can count by the number of inverse descents, or the number of inverse peaks, etc.

63 These homomorphisms may be defined more naturally on a larger algebra than the noncommutative symmetric functions, the Malvenuto-Reutenauer algebra (also called the algebra of free quasi-symmetric functions).

64 These homomorphisms may be defined more naturally on a larger algebra than the noncommutative symmetric functions, the Malvenuto-Reutenauer algebra (also called the algebra of free quasi-symmetric functions). To define it, we first define the standardization st(w) of the word w to be the permutation obtained by replacing the smallest entry by 1, the next smallest by 2, and so on, where repeated letters are ordered from left to right, so for example st(115212)=

65 Then for each 2 S n we define a series [ ] in the noncommuting variables X 1, X 2,... by [ ] = X X i1 X in st(i 1 i n)= For example, and [132] = X X a X c X b aappleb<c [12 n] =h n.

66 The Malvenuto-Reutenauer algebra is spanned by the [ ] with 2 S n for some n.

67 The Malvenuto-Reutenauer algebra is spanned by the [ ] with 2 S n for some n. It s easy to describe how these basis elements multiply: if 2 S m and 2 S n then [ ][ ]= X [ ] where the sum is over all = 1 m+n 2 S m+n for which st( 1 m )= and st( m+1 m+n )=.

68 Then the algebra of noncommutative symmetric functions is the subalgebra of the Malvenuto-Reutenauer algebra generating by the h n =[1 n]. Then the homomorphism MR, we can define either 1 extends in two different ways to or where 2 S n. x n 1a([ ]) = q inv( ) (q) n x n 1b([ ]) = q imaj( ), (q) n

69 We have the following relations between the algebras that we ve discussed: NCSF! MR?? y y SF! QSF