A Generalization of the Euler-Glaisher Bijection

Transcription

1 A Generalization of the Euler-Glaisher Bijection p.1/48 A Generalization of the Euler-Glaisher Bijection Andrew Sills Georgia Southern University

2 A Generalization of the Euler-Glaisher Bijection p.2/48 joint work with James Sellers and Gary Mullen Penn State University Electronic Journal of Combinatorics 11(1) (2004) #R43

3 PARTITION definition A Generalization of the Euler-Glaisher Bijection p.3/48 A partition λ of the integer n is a representation of n as an unordered sum of positive integers λ 1 + λ λ r = n.

4 PARTITION definition A Generalization of the Euler-Glaisher Bijection p.3/48 A partition λ of the integer n is a representation of n as an unordered sum of positive integers λ 1 + λ λ r = n. Each summand λ i is called a part of the partition λ.

5 PARTITION definition A Generalization of the Euler-Glaisher Bijection p.3/48 A partition λ of the integer n is a representation of n as an unordered sum of positive integers λ 1 + λ λ r = n. Each summand λ i is called a part of the partition λ. Often a canonical ordering of parts is imposed: λ 1 λ 2 λ r.

6 The Partitions of 6 A Generalization of the Euler-Glaisher Bijection p.4/

7 Euler s partition identity A Generalization of the Euler-Glaisher Bijection p.5/48

8 Euler s partition identity A Generalization of the Euler-Glaisher Bijection p.6/48 The number of partitions of n into odd parts equals the number of partitions of n into distinct parts.

9 Euler s partition identity Example A Generalization of the Euler-Glaisher Bijection p.7/48 Of the eleven partitions of 6, four of them have only odd parts:

10 Euler s partition identity Example A Generalization of the Euler-Glaisher Bijection p.7/48 Of the eleven partitions of 6, four of them have only odd parts: and four of them have distinct parts:

11 Frequency Notation for Partitions A Generalization of the Euler-Glaisher Bijection p.8/48 Any partition λ 1 + λ 2 + λ λ r may be written in the form f f f f ,

12 Frequency Notation for Partitions A Generalization of the Euler-Glaisher Bijection p.8/48 Any partition λ 1 + λ 2 + λ λ r may be written in the form or more briefly, as f f f f , {f 1,f 2,f 3,f 4,...}, where f i represents the number of appearances of the positive integer i in the partition.

13 Frequency Notation for Partitions A Generalization of the Euler-Glaisher Bijection p.9/48 For example, the partition

14 Frequency Notation for Partitions A Generalization of the Euler-Glaisher Bijection p.10/48 For example, the partition =

15 Frequency Notation for Partitions A Generalization of the Euler-Glaisher Bijection p.11/48 For example, the partition = may be represented by the frequency sequence {2, 4, 1, 2, 0, 4, 0, 0, 0, 0, 0, 0,... }.

16 A Generalization of the Euler-Glaisher Bijection p.12/48 Thus each sequence {f i } i=1, where each f i is a nonnegative integer and only finitely many of the f i are nonzero, represents a partition of the integer i=1 if i.

17 Glaisher s proof of Euler s identity A Generalization of the Euler-Glaisher Bijection p.13/48

18 Glaisher s proof of Euler s identity A Generalization of the Euler-Glaisher Bijection p.14/48 Let λ 1 + λ λ r be a partition λ of some positive integer n into r odd parts.

19 Glaisher s proof of Euler s identity A Generalization of the Euler-Glaisher Bijection p.14/48 Let λ 1 + λ λ r be a partition λ of some positive integer n into r odd parts. Rewrite λ in the form f f f

20 Glaisher s proof of Euler s identity A Generalization of the Euler-Glaisher Bijection p.14/48 Let λ 1 + λ λ r be a partition λ of some positive integer n into r odd parts. Rewrite λ in the form f f f Replace each f i with its binary expansion + a i3 8 + a i2 4 + a i1 2 + a i0 1.

21 Glaisher s proof of Euler s identity A Generalization of the Euler-Glaisher Bijection p.15/48 So, f f f f 7 7 +

22 Glaisher s proof of Euler s identity A Generalization of the Euler-Glaisher Bijection p.16/48 So, f f f f = ( + a 1,3 8 + a 1,2 4 + a 1,1 2 + a 1,0 1) 1 +( + a 3,3 8 + a 3,2 4 + a 3,1 2 + a 3,0 1) 3 +( + a 5,3 8 + a 5,2 4 + a 5,1 2 + a 5,0 1) 5.

23 Glaisher s proof of Euler s identity A Generalization of the Euler-Glaisher Bijection p.17/48 So, f f f f = ( + a 1,3 8 + a 1,2 4 + a 1,1 2 + a 1,0 1) 1 +( + a 3,3 8 + a 3,2 4 + a 3,1 2 + a 3,0 1) 3 +( + a 5,3 8 + a 5,2 4 + a 5,1 2 + a 5,0 1) 5. = a 1,0 + 2a 1,1 + 3a 3,0 + 4a 1,2 + 5a 5,0 + 6a 3,1 + 7a 7, where each a i,j {0, 1}.

24 Euler s partition identity A Generalization of the Euler-Glaisher Bijection p.18/48 The number of partitions of n into odd parts equals the number of partitions of n into distinct parts.

25 Euler s partition identity A Generalization of the Euler-Glaisher Bijection p.19/48 The number of partitions of n into nonmultiples of 2 equals the number of partitions of n where no part appears more than once.

26 Glaisher s partition identity A Generalization of the Euler-Glaisher Bijection p.20/48 The number of partitions of n into nonmultiples of m equals the number of partitions of n where no part appears more than m 1 times.

27 Proof of Glaisher s identity A Generalization of the Euler-Glaisher Bijection p.21/48

28 Proof of Glaisher s identity A Generalization of the Euler-Glaisher Bijection p.22/48 Let λ 1 + λ λ r be a partition λ of some positive integer n into r nonmultiples of m.

29 Proof of Glaisher s identity A Generalization of the Euler-Glaisher Bijection p.22/48 Let λ 1 + λ λ r be a partition λ of some positive integer n into r nonmultiples of m. Rewrite λ in the form where f i =0 if m i. f f f 3 3 +,

30 Proof of Glaisher s identity A Generalization of the Euler-Glaisher Bijection p.22/48 Let λ 1 + λ λ r be a partition λ of some positive integer n into r nonmultiples of m. Rewrite λ in the form where f i =0 if m i. f f f 3 3 +, Replace each f i with its base m expansion + a i3 m 3 + a i2 m 2 + a i1 m + a i0 1.

31 Proof of Glaisher s identity A Generalization of the Euler-Glaisher Bijection p.23/48 So, f f f f 4 4 +

32 Proof of Glaisher s identity A Generalization of the Euler-Glaisher Bijection p.24/48 So, f f f f = ( + a 1,3 m 3 + a 1,2 m 2 + a 1,1 m + a 1,0 1) 1 +( + a 2,3 m 3 + a 2,2 m 2 + a 2,1 m + a 2,0 1) 2 +( + a 3,3 m 3 + a 3,2 m 2 + a 3,1 m + a 3,0 1) 3. where each 0 a i,j m 1.

33 Partition Ideals A Generalization of the Euler-Glaisher Bijection p.25/48

34 Partition Ideals A Generalization of the Euler-Glaisher Bijection p.26/48 Informal Definition A partition ideal C is a set of partitions such that for each λ C, if one or more parts is removed from λ, the resulting partition is also in C.

35 The First Rogers-Ramanujan Identity A Generalization of the Euler-Glaisher Bijection p.27/48

36 The First Rogers-Ramanujan Identity A Generalization of the Euler-Glaisher Bijection p.28/48 Let R 1 (n) denote the number of partitions of n into parts congruent to ±1 (mod 5).

37 The First Rogers-Ramanujan Identity A Generalization of the Euler-Glaisher Bijection p.28/48 Let R 1 (n) denote the number of partitions of n into parts congruent to ±1 (mod 5). Let R 2 (n) denote the number of partitions λ λ r of n such that λ i λ i+1 2.

38 The First Rogers-Ramanujan Identity A Generalization of the Euler-Glaisher Bijection p.28/48 Let R 1 (n) denote the number of partitions of n into parts congruent to ±1 (mod 5). Let R 2 (n) denote the number of partitions λ λ r of n such that λ i λ i+1 2. Then R 1 (n) = R 2 (n) for all n.

39 The First Rogers-Ramanujan Identity A Generalization of the Euler-Glaisher Bijection p.29/48

40 The First Rogers-Ramanujan Identity A Generalization of the Euler-Glaisher Bijection p.30/48 The partitions enumerated by R 1 (n) are those for which f i = 0 whenever i ±1 (mod 5).

41 The First Rogers-Ramanujan Identity A Generalization of the Euler-Glaisher Bijection p.30/48 The partitions enumerated by R 1 (n) are those for which f i = 0 whenever i ±1 (mod 5). The partitions enumerated by R 2 (n) are those for which f i + f i+1 1.

42 Minimal Bounding Sequence A Generalization of the Euler-Glaisher Bijection p.31/48 Let the sequence {d C 1,d C 2,d C 3,...} be defined by d C j = sup f j, {f i } i=1 C

43 Minimal Bounding Sequence A Generalization of the Euler-Glaisher Bijection p.31/48 Let the sequence {d C 1,d C 2,d C 3,...} be defined by d C j = sup f j, {f i } i=1 C each d i is a nonnegative integer or +.

44 Euler s Theorem Revisited A Generalization of the Euler-Glaisher Bijection p.32/48 Let O denote the set of all partitions with only odd parts.

45 Euler s Theorem Revisited A Generalization of the Euler-Glaisher Bijection p.32/48 Let O denote the set of all partitions with only odd parts. {d O j } j=1 = {, 0,, 0,, 0,... }

46 Euler s Theorem Revisited A Generalization of the Euler-Glaisher Bijection p.32/48 Let O denote the set of all partitions with only odd parts. {d O j } j=1 = {, 0,, 0,, 0,... } Let D denote the set of all partitions with distinct parts.

47 Euler s Theorem Revisited A Generalization of the Euler-Glaisher Bijection p.32/48 Let O denote the set of all partitions with only odd parts. {d O j } j=1 = {, 0,, 0,, 0,... } Let D denote the set of all partitions with distinct parts. {d D j } j=1 = {1, 1, 1, 1, 1, 1,... }.

48 Equivalent Partition Ideals A Generalization of the Euler-Glaisher Bijection p.33/48 Let p(c,n) denote the number of partitions of an integer n in the partition ideal C.

49 Equivalent Partition Ideals A Generalization of the Euler-Glaisher Bijection p.33/48 Let p(c,n) denote the number of partitions of an integer n in the partition ideal C. We say that two partition ideals C 1 and C 2 are equivalent, and write C 1 C 2, if p(c 1,n) = p(c 2,n) for all integers n.

50 Euler s Partition Identity A Generalization of the Euler-Glaisher Bijection p.34/48 O D.

51 The Multiset Associated with a Partition Ideal of Order 1 A Generalization of the Euler-Glaisher Bijection p.35/48 Define the multiset associated with C, M(C), as follows: M(C) := {j(d C j + 1) j Z + and d C j < }.

52 The Multiset Associated with a Partition Ideal of Order 1 A Generalization of the Euler-Glaisher Bijection p.35/48 Define the multiset associated with C, M(C), as follows: M(C) := {j(d C j + 1) j Z + and d C j < }. Andrews proved C 1 C 2 if and only if M(C 1 ) = M(C 2 ).

53 The Multiset Associated with a Partition Ideal of Order 1 A Generalization of the Euler-Glaisher Bijection p.36/48 M(O) = M(D) = {2, 4, 6, 8, 10, 12,... }

54 Partitions of Infinity A Generalization of the Euler-Glaisher Bijection p.37/48 MacMahon defined a partition of infinity to be a formal expression of the form (g 1 1) 1+(g 2 1) g 1 +(g 3 1) (g 1 g 2 )+(g 4 1) (g 1 g 2 g 3 )+... where each g i 2 or for some fixed k, g 1,g 2,g 3,...,g k 1 > 1, g k =, and g k+1 = g k+2 = g k+3 = = 1.

55 Partitions of Infinity A Generalization of the Euler-Glaisher Bijection p.38/48 Note that a partition of infinity may be thought of as a minimal bounding sequence for a partition ideal of order one with d 1 = g 1 1 d g1 = g 2 1 d g1 g 2 = g 3 1 d g1 g 2 g 3 = g 4 1. and d i = 0 if i {1,g 1,g 1 g 2,g 1 g 2 g 3,...}.

56 Partitions of Infinity A Generalization of the Euler-Glaisher Bijection p.39/48 If g i = m for all i, we have a minimal bounding sequence for the partition ideal of the base m expansion.

57 A Generalization of the Euler-Glaisher Bijection p.40/48 All partitions of infinity have generating function 1 1 q.

58 Given any partition ideal of order 1 C for which each term in the minimal bounding sequence is 0 or, A Generalization of the Euler-Glaisher Bijection p.41/48

59 A Generalization of the Euler-Glaisher Bijection p.41/48 Given any partition ideal of order 1 C for which each term in the minimal bounding sequence is 0 or, and any equivalent partition ideal of order 1 C,

60 A Generalization of the Euler-Glaisher Bijection p.41/48 Given any partition ideal of order 1 C for which each term in the minimal bounding sequence is 0 or, and any equivalent partition ideal of order 1 C, there exists a collection of partitions of infinity which gives rise to a Glaisher-type bijection from C to C.

61 A Generalization of the Euler-Glaisher Bijection p.41/48 Given any partition ideal of order 1 C for which each term in the minimal bounding sequence is 0 or, and any equivalent partition ideal of order 1 C, there exists a collection of partitions of infinity which gives rise to a Glaisher-type bijection from C to C. Further, there is an explicit algorithm for finding the required partitions of infinity.

62 An Inelegant Partition Identity A Generalization of the Euler-Glaisher Bijection p.42/48 Let C denote the set of partitions into parts not equal to 2, 9, 10, 12, 18 or 20.

63 An Inelegant Partition Identity A Generalization of the Euler-Glaisher Bijection p.42/48 Let C denote the set of partitions into parts not equal to 2, 9, 10, 12, 18 or 20. Let C denote the set of partitions where the parts 1, 9, and 10 may appear at most once, 3 and 4 may appear at most twice, 2 may appear at most four times, and all other positive integers may appear without restriction.

64 An Inelegant Partition Identity A Generalization of the Euler-Glaisher Bijection p.42/48 Let C denote the set of partitions into parts not equal to 2, 9, 10, 12, 18 or 20. Let C denote the set of partitions where the parts 1, 9, and 10 may appear at most once, 3 and 4 may appear at most twice, 2 may appear at most four times, and all other positive integers may appear without restriction. C C.

65 An Inelegant Partition Identity A Generalization of the Euler-Glaisher Bijection p.43/48 d C j = { 0 if j {2, 9, 10, 12, 18, 20} otherwise. {d C j } j=1 = {1, 4, 2, 2,,,,, 1, 1,,,,,... }.

66 An Inelegant Partition Identity A Generalization of the Euler-Glaisher Bijection p.43/48 d C j = { 0 if j {2, 9, 10, 12, 18, 20} otherwise. {d C j } j=1 = {1, 4, 2, 2,,,,, 1, 1,,,,,... }. Any partition of n in C can be written in the form 8 n = f f i i + f i=3 +f f i i, i=21 17 i=13 f i i where each f i is a nonnegative integer.

67 An Inelegant Partition Identity A Generalization of the Euler-Glaisher Bijection p.44/48 Expand f 1 by the partition of infinity defined by g 1,1 = 2, g 1,2 = 5, g 1,3 = 2, g 1,4 =, g 1,k = 1 if k > 4.

68 An Inelegant Partition Identity A Generalization of the Euler-Glaisher Bijection p.44/48 Expand f 1 by the partition of infinity defined by g 1,1 = 2, g 1,2 = 5, g 1,3 = 2, g 1,4 =, g 1,k = 1 if k > 4. Expand f 3 by the partition of infinity defined by g 3,1 = 3, g 3,2 = 2, g 3,3 =, g 3,k = 1 if k > 3.

69 An Inelegant Partition Identity A Generalization of the Euler-Glaisher Bijection p.44/48 Expand f 1 by the partition of infinity defined by g 1,1 = 2, g 1,2 = 5, g 1,3 = 2, g 1,4 =, g 1,k = 1 if k > 4. Expand f 3 by the partition of infinity defined by g 3,1 = 3, g 3,2 = 2, g 3,3 =, g 3,k = 1 if k > 3. Expand f 4 by the partition of infinity defined by g 4,1 = 3, g 4,2 =, g 4,k = 1 if k > 2.

70 n = ( a1,0 (1) + a 1,1 (2) + a 1,2 (2 5) + a 1,3 (2 5 2) ) 1 + ( a 2,0 (1) + a 2,1 (3) + a 2,2 (3 2) ) 3 + ( a 4,0 (1) + a 4,1 (3) ) 4 + ( a 5,0 (1) ) 5 + ( a 6,0 (1) ) 6 + ( a 7,0 (1) ) 7 + ( a 8,0 (1) ) 8 + ( a 11,0 (1) ) 11. where 0 a j,k g j,k+1 1 = d jgj,1 g j,2...g j,k. A Generalization of the Euler-Glaisher Bijection p.45/48

71 A Generalization of the Euler-Glaisher Bijection p.46/48 Apply the distributive property to obtain n = a 1,0 (1) + a 1,1 (2) + a 1,2 (10) + a 1,4 (20) + a 2,0 (3) + a 2,1 (9) + a 2,2 (18) + a 4,0 (4) + a 4,1 (12) + a 5,0 (5) + a 6,0 (6) where, in particular,. a 1,0 1, a 1,1 4, a 1,2 2, a 2,0 2, a 2,1 1, a 4,0 2.

72 Canonical Bijections A Generalization of the Euler-Glaisher Bijection p.47/48 Any C for which each term in {d C j } j=1 an M(C) with no repeated elements. is 0 or has

73 Canonical Bijections A Generalization of the Euler-Glaisher Bijection p.47/48 Any C for which each term in {d C j } j=1 an M(C) with no repeated elements. is 0 or has If for some C, M(C ) has no repeated elements, it is equivalent to some C with minimal bounding sequence consisting of only 0 s and s.

74 Canonical Bijections A Generalization of the Euler-Glaisher Bijection p.47/48 Any C for which each term in {d C j } j=1 an M(C) with no repeated elements. is 0 or has If for some C, M(C ) has no repeated elements, it is equivalent to some C with minimal bounding sequence consisting of only 0 s and s. This C maps bijectively to C via the Glaisher-type bijection where the role of the base m expansion is played by an appropriate partition of infinity.

75 Canonical Bijections A Generalization of the Euler-Glaisher Bijection p.47/48 Any C for which each term in {d C j } j=1 an M(C) with no repeated elements. is 0 or has If for some C, M(C ) has no repeated elements, it is equivalent to some C with minimal bounding sequence consisting of only 0 s and s. This C maps bijectively to C via the Glaisher-type bijection where the role of the base m expansion is played by an appropriate partition of infinity. If C C and M(C ) = M(C ) has no repeated elements, there is a canonical bijection between them.

76 Further Canonical Bijections for Equivalent Partition Ideals A Generalization of the Euler-Glaisher Bijection p.48/48 Max Kanovich Professor of Computer Science Queen Mary, University of London