Integer Partitions From A Geometric Viewpoint Matthias Beck Nguyen - PowerPoint PPT Presentation

Integer Partitions From A Geometric Viewpoint Matthias Beck Nguyen Le San Francisco State University of New South Wales Thomas Bliem Sunyoung Lee Benjamin Braun Carla Savage University of Kentucky NC State Ira Gessel Zafeirakis Zafeirakopoulos Brandeis University RISC Linz Matthias K¨ oppe UC Davis Thanks to: Ramanujan Journal (2010), arXiv:0906.5573 AIM Ehrenpreis memorial volume (2012), arXiv:1103.1070 NSF Journal of Algebraic Combinatorics (2013), arXiv:1206.1551 Ramanujan Journal (to appear), arXiv:1211.0258

“If things are nice there is probably a good reason why they are nice: and if you do not know at least one reason for this good fortune, then you still have work to do.” Richard Askey (Ramanujan and Important Formulas, Srinivasa Ramanujan (1887–1920), a Tribute , K. R. Nagarajan and T. Soundarajan, eds., Madurai Kamaraj University, 1987.) Integer Partitions From A Geometric Viewpoint Matthias Beck 2

“If things are nice there is probably a good reason why they are nice: and if you do not know at least one reason for this good fortune, then you still have work to do.” Richard Askey (Ramanujan and Important Formulas, Srinivasa Ramanujan (1887–1920), a Tribute , K. R. Nagarajan and T. Soundarajan, eds., Madurai Kamaraj University, 1987.) Polyhedral Geometry Partition Analysis Arithmetic Integer Partitions From A Geometric Viewpoint Matthias Beck 2

Integer Partitions A partition λ = ( λ 1 , λ 2 , . . . , λ n ) ∈ Z n of an integer k ≥ 0 satisfies k = λ 1 + λ 2 + · · · + λ n and 0 ≤ λ 1 ≤ λ 2 ≤ · · · ≤ λ n Example 5 = 1 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 2 = 1 + 2 + 2 = 1 + 1 + 3 = 2 + 3 = 1 + 4 = 5 Integer Partitions From A Geometric Viewpoint Matthias Beck 3

Integer Partitions A partition λ = ( λ 1 , λ 2 , . . . , λ n ) ∈ Z n of an integer k ≥ 0 satisfies k = λ 1 + λ 2 + · · · + λ n and 0 ≤ λ 1 ≤ λ 2 ≤ · · · ≤ λ n Number Theory ◮ Combinatorics ◮ Symmetric functions ◮ Representation Theory ◮ Physics ◮ Integer Partitions From A Geometric Viewpoint Matthias Beck 3

Integer Partitions A partition λ = ( λ 1 , λ 2 , . . . , λ n ) of an integer k ≥ 0 satisfies k = λ 1 + λ 2 + · · · + λ n and 0 ≤ λ 1 ≤ λ 2 ≤ · · · ≤ λ n � q λ 1 + ··· + λ n Goal Compute λ where the sum runs through your favorite partitions. Integer Partitions From A Geometric Viewpoint Matthias Beck 4

Integer Partitions A partition λ = ( λ 1 , λ 2 , . . . , λ n ) of an integer k ≥ 0 satisfies k = λ 1 + λ 2 + · · · + λ n and 0 ≤ λ 1 ≤ λ 2 ≤ · · · ≤ λ n � q λ 1 + ··· + λ n Goal Compute λ where the sum runs through your favorite partitions. Example (Euler’s mother-of-all-partition-identities) # partitions of k into odd parts = # partitions of k into distinct parts Integer Partitions From A Geometric Viewpoint Matthias Beck 4

Integer Partitions A partition λ = ( λ 1 , λ 2 , . . . , λ n ) of an integer k ≥ 0 satisfies k = λ 1 + λ 2 + · · · + λ n and 0 ≤ λ 1 ≤ λ 2 ≤ · · · ≤ λ n � q λ 1 + ··· + λ n Goal Compute λ where the sum runs through your favorite partitions. Example (triangle partitions) T := { λ : 1 ≤ λ 1 ≤ λ 2 ≤ λ 3 , λ 1 + λ 2 > λ 3 } q 3 q λ 1 + λ 2 + λ 3 = � (1 − q 2 )(1 − q 3 )(1 − q 4 ) λ ∈ T Integer Partitions From A Geometric Viewpoint Matthias Beck 4

Integer Partitions A partition λ = ( λ 1 , λ 2 , . . . , λ n ) of an integer k ≥ 0 satisfies k = λ 1 + λ 2 + · · · + λ n and 0 ≤ λ 1 ≤ λ 2 ≤ · · · ≤ λ n � q λ 1 + ··· + λ n Goal Compute λ where the sum runs through your favorite partitions. Example (triangle partitions) T := { λ : 1 ≤ λ 1 ≤ λ 2 ≤ λ 3 , λ 1 + λ 2 > λ 3 } q 3 q λ 1 + λ 2 + λ 3 = � (1 − q 2 )(1 − q 3 )(1 − q 4 ) λ ∈ T � k 3 � � k � � k + 2 � − → # partitions of k in T equals − 12 4 4 Integer Partitions From A Geometric Viewpoint Matthias Beck 4

n -gon Partitions P n := { λ : 1 ≤ λ 1 ≤ λ 2 ≤ · · · ≤ λ n , λ 1 + · · · + λ n − 1 > λ n } (Sample) Theorem 1 (Andrews, Paule & Riese 2001) q q λ 1 + ··· + λ n = � (1 − q )(1 − q 2 ) · · · (1 − q n ) λ ∈ P n q 2 n − 2 − (1 − q )(1 − q 2 )(1 − q 4 ) · · · (1 − q 2 n − 2 ) Integer Partitions From A Geometric Viewpoint Matthias Beck 5

n -gon Partitions P n := { λ : 1 ≤ λ 1 ≤ λ 2 ≤ · · · ≤ λ n , λ 1 + · · · + λ n − 1 > λ n } (Sample) Theorem 1 (Andrews, Paule & Riese 2001) q q λ 1 + ··· + λ n = � (1 − q )(1 − q 2 ) · · · (1 − q n ) λ ∈ P n q 2 n − 2 − (1 − q )(1 − q 2 )(1 − q 4 ) · · · (1 − q 2 n − 2 ) Natural extension: symmetrize, e.g., the triangle condition to λ π (1) + λ π (2) > λ π (3) ∀ π ∈ S 3 and enumerate compositions λ with this condition. Integer Partitions From A Geometric Viewpoint Matthias Beck 5

Symmetrically Constrained Compositions (Sample) Theorem 2 (Andrews, Paule & Riese 2001) Given positive integers b and n ≥ 2 , let K consist of all nonnegative integer sequences λ satisfying b ( λ π (1) + · · · + λ π ( n − 1) ) ≥ ( nb − b − 1) λ π ( n ) ∀ π ∈ S n 1 − q n ( nb − 1) q λ 1 + ··· + λ n = � Then (1 − q n )(1 − q nb − 1 ) n λ ∈ K Integer Partitions From A Geometric Viewpoint Matthias Beck 6

Symmetrically Constrained Compositions (Sample) Theorem 2 (Andrews, Paule & Riese 2001) Given positive integers b and n ≥ 2 , let K consist of all nonnegative integer sequences λ satisfying b ( λ π (1) + · · · + λ π ( n − 1) ) ≥ ( nb − b − 1) λ π ( n ) ∀ π ∈ S n 1 − q n ( nb − 1) q λ 1 + ··· + λ n = � Then (1 − q n )(1 − q nb − 1 ) n λ ∈ K Andrews, Paule & Riese found several identities of this form; all of them concerned symmetric constraints of the form a 1 λ π (1) + a 2 λ π (2) + · · · + a n λ π ( n ) ≥ 0 ∀ π ∈ S n with the condition a 1 + · · · + a n = 1 . Integer Partitions From A Geometric Viewpoint Matthias Beck 6

Enter Geometry We view a partition λ = ( λ 1 , λ 2 , . . . , λ n ) as an integer lattice point in (a subcone of) { x ∈ R n : 0 ≤ x 1 ≤ x 2 ≤ · · · ≤ x n } n � R ≥ 0 v j is unimodular if v 1 , . . . , v n form a lattice basis of Z n C = j =1 1 x m = � − → σ C ( x ) := � n j =1 (1 − x v j ) m ∈ C ∩ Z n where x m := x m 1 · · · x m n 1 n Integer Partitions From A Geometric Viewpoint Matthias Beck 7

Enter Geometry We view a partition λ = ( λ 1 , λ 2 , . . . , λ n ) as an integer lattice point in (a subcone of) { x ∈ R n : 0 ≤ x 1 ≤ x 2 ≤ · · · ≤ x n } n � R ≥ 0 v j is unimodular if v 1 , . . . , v n form a lattice basis of Z n C = j =1 1 x m = j =1 (1 − x v j ) where x m := x m 1 � · · · x m n − → σ C ( x ) := � n 1 n m ∈ C ∩ Z n Example P := { x ∈ R n : 0 ≤ x 1 ≤ x 2 ≤ · · · ≤ x n } is unimodular with generators e n , e n − 1 + e n , . . . , e 1 + e 2 + · · · + e n Integer Partitions From A Geometric Viewpoint Matthias Beck 7

Enter Geometry We view a partition λ = ( λ 1 , λ 2 , . . . , λ n ) as an integer lattice point in (a subcone of) { x ∈ R n : 0 ≤ x 1 ≤ x 2 ≤ · · · ≤ x n } n � R ≥ 0 v j is unimodular if v 1 , . . . , v n form a lattice basis of Z n C = j =1 1 x m = j =1 (1 − x v j ) where x m := x m 1 � · · · x m n − → σ C ( x ) := � n 1 n m ∈ C ∩ Z n Remark This geometric viewpoint is not new: Pak (Proceedings AMS 2004, Ramanujan Journal 2006) realized that several partition identities can be interpreted as bijections of lattice points in two unimodular cones. Corteel, Savage & Wilf (Integers 2005) discussed several families of partitions/compositions giving rise to unimodular cones (and thus a nice product description of their generating function). Integer Partitions From A Geometric Viewpoint Matthias Beck 7

n -gon Partitions Revisited Theorem 1 (Andrews, Paule & Riese 2001) q q λ 1 + ··· + λ n = � (1 − q )(1 − q 2 ) · · · (1 − q n ) λ ∈ P n q 2 n − 2 − (1 − q )(1 − q 2 )(1 − q 4 ) · · · (1 − q 2 n − 2 ) An n -gon partition λ ∈ P n lies in the “fat” cone C 1 := { x ∈ R n : 0 < x 1 ≤ x 2 ≤ · · · ≤ x n , x 1 + · · · + x n − 1 > x n } Integer Partitions From A Geometric Viewpoint Matthias Beck 8

n -gon Partitions Revisited x = x 1 2 x = 0 1 C 2 Theorem 1 (Andrews, Paule & Riese 2001) x + x = x 2 3 1 q q λ 1 + ··· + λ n = � C 1 (1 − q )(1 − q 2 ) · · · (1 − q n ) λ ∈ P n q 2 n − 2 − (1 − q )(1 − q 2 )(1 − q 4 ) · · · (1 − q 2 n − 2 ) x = x 2 3 An n -gon partition λ ∈ P n lies in the “fat” cone C 1 := { x ∈ R n : 0 < x 1 ≤ x 2 ≤ · · · ≤ x n , x 1 + · · · + x n − 1 > x n } However, C 1 = P \ C 2 for the unimodular cone C 2 := { x ∈ R n : 0 < x 1 ≤ x 2 ≤ · · · ≤ x n , x 1 + · · · + x n − 1 ≤ x n } Theorem 1 is the statement σ C 1 ( q, . . . , q ) = σ P ( q, . . . , q ) − σ C 2 ( q, . . . , q ) Integer Partitions From A Geometric Viewpoint Matthias Beck 8