Congruences for Fishburn numbers modulo prime powers Partitions, q - PowerPoint PPT Presentation

Congruences for Fishburn numbers modulo prime powers Partitions, q -series, and modular forms AMS Joint Mathematics Meetings, San Antonio Armin Straub January 11, 2015 University of Illinois at Urbana–Champaign 3 2 1 1 1 1 ξ (3) = 5 1 2 1 1 1 Fishburn matrices of size 3 Congruences for Fishburn numbers modulo prime powers Armin Straub 1 / 16

Examples of partitions • The integer partitions of 3 : p (3) = 3 Congruences for Fishburn numbers modulo prime powers Armin Straub 2 / 16

Examples of partitions • The integer partitions of 3 : p (3) = 3 The Fishburn matrices of size 3 : • ξ (3) = 5 3 2 1 1 1 1 1 2 1 1 1 A Fishburn matrix is an DEF • upper-triangular matrix with entries in Z � 0 , such that • every row and column contains at least one non-zero entry. Its size is the sum of the entries. Congruences for Fishburn numbers modulo prime powers Armin Straub 2 / 16

Examples of partitions The integer partitions of 4 : • p (4) = 5 The Fishburn matrices of size 4 : • 4 3 2 2 1 1 1 1 1 2 ξ (4) = 15 1 2 1 2 3 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 Congruences for Fishburn numbers modulo prime powers Armin Straub 3 / 16

Examples of partitions The integer partitions of 4 : • p (4) = 5 The Fishburn matrices of size 4 : • 4 3 2 2 1 1 1 1 1 2 ξ (4) = 15 1 2 1 2 3 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 primitive Fishburn 1 1 matrices 1 1 ξ − 1 (4) = 5 1 Congruences for Fishburn numbers modulo prime powers Armin Straub 3 / 16

Asymptotic facts p ( n ) : 1 , 1 , 2 , 3 , 5 , 7 , 11 , 15 , 22 , 30 , 42 , . . . p (2015) ≈ 7 . 20 × 10 45 ξ ( n ) : 1 , 1 , 2 , 5 , 15 , 53 , 217 , 1014 , 5335 , 31240 , 201608 , . . . ξ (2015) ≈ 4 . 05 × 10 5351 Congruences for Fishburn numbers modulo prime powers Armin Straub 4 / 16

Asymptotic facts p ( n ) : 1 , 1 , 2 , 3 , 5 , 7 , 11 , 15 , 22 , 30 , 42 , . . . p (2015) ≈ 7 . 20 × 10 45 ξ ( n ) : 1 , 1 , 2 , 5 , 15 , 53 , 217 , 1014 , 5335 , 31240 , 201608 , . . . ξ (2015) ≈ 4 . 05 × 10 5351 3 e π √ 1 THM 2 n/ 3 p ( n ) ∼ √ Hardy– Ramanujan 4 n 1918 √ � 6 THM � n ξ ( n ) ∼ 12 3 n π 5 / 2 e π 2 / 12 Zagier n ! π 2 2001 Congruences for Fishburn numbers modulo prime powers Armin Straub 4 / 16

Asymptotic facts p ( n ) : 1 , 1 , 2 , 3 , 5 , 7 , 11 , 15 , 22 , 30 , 42 , . . . p (2015) ≈ 7 . 20 × 10 45 ξ ( n ) : 1 , 1 , 2 , 5 , 15 , 53 , 217 , 1014 , 5335 , 31240 , 201608 , . . . ξ (2015) ≈ 4 . 05 × 10 5351 3 e π √ 1 THM 2 n/ 3 p ( n ) ∼ √ Hardy– Ramanujan 4 n 1918 √ � 6 THM � n ξ ( n ) ∼ 12 3 n π 5 / 2 e π 2 / 12 Zagier n ! π 2 2001 • Primitive Fishburn matrices are those with entries 0 , 1 only. # of primitive Fishburn matrices of size n # of Fishburn matrices of size n ( = ξ ( n ) ) = e − π 2 / 6 ≈ 0 . 193 lim n →∞ Jel´ ınek–Drmota, 2011 Congruences for Fishburn numbers modulo prime powers Armin Straub 4 / 16

Fishburn numbers • The Fishburn numbers ξ ( n ) have the following generating function. THM n ξ ( n ) q n = � � � (1 − (1 − q ) j ) = F (1 − q ) Zagier 2001 n � 0 n � 0 j =1 • F ( q ) is Kontsevich’s “strange” function (more later) Congruences for Fishburn numbers modulo prime powers Armin Straub 5 / 16

Fishburn numbers • The Fishburn numbers ξ ( n ) have the following generating function. THM n ξ ( n ) q n = � � � (1 − (1 − q ) j ) = F (1 − q ) Zagier 2001 n � 0 n � 0 j =1 • F ( q ) is Kontsevich’s “strange” function (more later) 1 • The primitive Fishburn numbers have generating function F ( 1+ q ) . Congruences for Fishburn numbers modulo prime powers Armin Straub 5 / 16

Fishburn numbers • The Fishburn numbers ξ ( n ) have the following generating function. THM n ξ ( n ) q n = � � � (1 − (1 − q ) j ) = F (1 − q ) Zagier 2001 n � 0 n � 0 j =1 • F ( q ) is Kontsevich’s “strange” function (more later) 1 • The primitive Fishburn numbers have generating function F ( 1+ q ) . • Garvan introduces the numbers ξ r,s ( n ) by n ξ r,s ( n ) q n = (1 − q ) s � � � (1 − (1 − q ) rj ) = (1 − q ) s F ((1 − q ) r ) . n � 0 n � 0 j =1 • ξ ( n ) = ξ 1 , 0 ( n ) • ( − 1) n ξ − 1 , 0 ( n ) count primitive Fishburn matrices • For instance, ξ 1 , 3 ( n ) = ξ ( n ) − 3 ξ ( n − 1) + 3 ξ ( n − 2) − ξ ( n − 3) . Congruences for Fishburn numbers modulo prime powers Armin Straub 5 / 16

Interval orders and Fishburn matrices • An interval order is a poset consisting of intervals I ⊆ R with order given by: I < J ⇐ ⇒ i < j for all i ∈ I , j ∈ J 1 2 3 R 1 3 2 5 4 Congruences for Fishburn numbers modulo prime powers Armin Straub 6 / 16

Interval orders and Fishburn matrices • An interval order is a poset consisting of intervals I ⊆ R with order given by: I < J ⇐ ⇒ i < j for all i ∈ I , j ∈ J FACT ξ ( n ) = # of interval orders of size n (up to isomorphism) 1 2 3 R 1 3 2 5 4 Congruences for Fishburn numbers modulo prime powers Armin Straub 6 / 16

Interval orders and Fishburn matrices • An interval order is a poset consisting of intervals I ⊆ R with order given by: I < J ⇐ ⇒ i < j for all i ∈ I , j ∈ J FACT ξ ( n ) = # of interval orders of size n (up to isomorphism) 1 2 3 R 1 3 2 5 4 ↓ ↓ standard representation 1 2 3 R 1 3 2 5 4 Congruences for Fishburn numbers modulo prime powers Armin Straub 6 / 16

Interval orders and Fishburn matrices • An interval order is a poset consisting of intervals I ⊆ R with order given by: I < J ⇐ ⇒ i < j for all i ∈ I , j ∈ J FACT ξ ( n ) = # of interval orders of size n (up to isomorphism) 1 2 3 R Fishburn matrix M 1 3 2 5 4 M i,j = # of intervals [ i, j ] ↓ ↓ standard representation 1 1 2 1 2 3 1 R 1 3 2 5 4 Congruences for Fishburn numbers modulo prime powers Armin Straub 6 / 16

Partition congruences • Ramanujan proved the striking congruences p (5 m − 1) ≡ 0 (mod 5) , p (7 m − 2) ≡ 0 (mod 7) , p (11 m − 5) ≡ 0 (mod 11) . Congruences for Fishburn numbers modulo prime powers Armin Straub 7 / 16

Partition congruences • Ramanujan proved the striking congruences p (5 m − 1) ≡ 0 (mod 5) , p (7 m − 2) ≡ 0 (mod 7) , p (11 m − 5) ≡ 0 (mod 11) . • Also conjectured generalizations with moduli powers of 5 , 7 , 11 . p (5 λ m − δ 5 ( λ )) ≡ 0 (mod 5 λ ) , p (7 λ m − δ 7 ( λ )) ≡ 0 (mod 7 λ ) , p (11 λ m − δ 11 ( λ )) ≡ 0 (mod 11 λ ) , where δ p ( λ ) ≡ − 1 / 24 (mod p λ ) . Congruences for Fishburn numbers modulo prime powers Armin Straub 7 / 16

Partition congruences • Ramanujan proved the striking congruences p (5 m − 1) ≡ 0 (mod 5) , p (7 m − 2) ≡ 0 (mod 7) , p (11 m − 5) ≡ 0 (mod 11) . • Also conjectured generalizations with moduli powers of 5 , 7 , 11 . p (5 λ m − δ 5 ( λ )) ≡ 0 (mod 5 λ ) , p (7 λ m − δ 7 ( λ )) ≡ 0 (mod 7 ⌊ λ/ 2 ⌋ +1 ) , p (11 λ m − δ 11 ( λ )) ≡ 0 (mod 11 λ ) , where δ p ( λ ) ≡ − 1 / 24 (mod p λ ) . Watson (1938), Atkin (1967) Congruences for Fishburn numbers modulo prime powers Armin Straub 7 / 16

Partition congruences • Ramanujan proved the striking congruences p (5 m − 1) ≡ 0 (mod 5) , p (7 m − 2) ≡ 0 (mod 7) , p (11 m − 5) ≡ 0 (mod 11) . • Also conjectured generalizations with moduli powers of 5 , 7 , 11 . p (5 λ m − δ 5 ( λ )) ≡ 0 (mod 5 λ ) , p (7 λ m − δ 7 ( λ )) ≡ 0 (mod 7 ⌊ λ/ 2 ⌋ +1 ) , p (11 λ m − δ 11 ( λ )) ≡ 0 (mod 11 λ ) , where δ p ( λ ) ≡ − 1 / 24 (mod p λ ) . Watson (1938), Atkin (1967) There appear to be additional congruences, such as RK p (7 2 m − 2 , 9 , 16 , 30) ≡ 0 (mod 7 2 ) , p (5 3 m − 1 , 26 , 51) ≡ 0 (mod 5 3 ) . Congruences for Fishburn numbers modulo prime powers Armin Straub 7 / 16

The Andrews–Sellers congruences Let p be a prime, and j ∈ Z > 0 such that THM Andrews, Sellers � 1 − 24 k � 2014 = − 1 for k = 1 , 2 , . . . , j . (AS) p Then, for all m � 1 , ξ ( pm − j ) ≡ 0 (mod p ) . EG ξ (5 m − 1) ≡ ξ (5 m − 2) ≡ 0 (mod 5) ξ (7 m − 1) ≡ 0 (mod 7) ξ (11 m − 1) ≡ ξ (11 m − 2) ≡ ξ (11 m − 3) ≡ 0 (mod 11) Congruences for Fishburn numbers modulo prime powers Armin Straub 8 / 16

The Andrews–Sellers congruences Let p be a prime, and j ∈ Z > 0 such that THM Andrews, Sellers � 1 − 24 k � 2014 = − 1 for k = 1 , 2 , . . . , j . (AS) p Then, for all m � 1 , ξ ( pm − j ) ≡ 0 (mod p ) . EG ξ (5 m − 1) ≡ ξ (5 m − 2) ≡ 0 (mod 5) ξ (7 m − 1) ≡ 0 (mod 7) ξ (11 m − 1) ≡ ξ (11 m − 2) ≡ ξ (11 m − 3) ≡ 0 (mod 11) • Garvan (2014) proved that (AS) may be replaced with � 1 − 24 k � � = 1 for k = 1 , 2 , . . . , j , p and that analogous congruences hold for the generalizations ξ r,s ( n ) . Congruences for Fishburn numbers modulo prime powers Armin Straub 8 / 16