Congruences for Fishburn numbers modulo prime powers

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1 Congruences for Fishburn numbers modulo prime powers Partitions, q-series, and modular forms AMS Joint Mathematics Meetings, San Antonio January, 205 University of Illinois at Urbana Champaign ξ(3) = Fishburn matrices of size 3 / 6

2 Examples of partitions The integer partitions of 3: p(3) = 3 2 / 6

3 Examples of partitions The integer partitions of 3: p(3) = 3 The Fishburn matrices of size 3: ξ(3) = DEF A Fishburn matrix is an 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. 2 / 6

4 Examples of partitions The integer partitions of 4: p(4) = 5 The Fishburn matrices of size 4: ξ(4) = / 6

5 Examples of partitions The integer partitions of 4: p(4) = 5 The Fishburn matrices of size 4: ξ(4) = primitive Fishburn matrices ξ (4) = 5 3 / 6

6 Asymptotic facts p(n) :,, 2, 3, 5, 7,, 5, 22, 30, 42,... p(205) ξ(n) :,, 2, 5, 5, 53, 27, 04, 5335, 3240, 20608,... ξ(205) / 6

7 Asymptotic facts p(n) :,, 2, 3, 5, 7,, 5, 22, 30, 42,... p(205) ξ(n) :,, 2, 5, 5, 53, 27, 04, 5335, 3240, 20608,... ξ(205) THM Hardy Ramanujan 98 THM Zagier 200 p(n) 4n 3 eπ 2n/3 ξ(n) 2 ( ) 3n /2 6 n π 5/2 eπ2 π 2 n! 4 / 6

8 Asymptotic facts p(n) :,, 2, 3, 5, 7,, 5, 22, 30, 42,... p(205) ξ(n) :,, 2, 5, 5, 53, 27, 04, 5335, 3240, 20608,... ξ(205) THM Hardy Ramanujan 98 THM Zagier 200 p(n) 4n 3 eπ 2n/3 ξ(n) 2 ( ) 3n /2 6 n π 5/2 eπ2 π 2 n! Primitive Fishburn matrices are those with entries 0, only. # of primitive Fishburn matrices of size n lim n # of Fishburn matrices of size n = /6 (= ξ(n)) e π Jeĺınek Drmota, 20 4 / 6

9 Fishburn numbers The Fishburn numbers ξ(n) have the following generating function. THM Zagier 200 n ( ( q) j ) = F ( q) n 0 ξ(n)q n = n 0 j= F (q) is Kontsevich s strange function (more later) 5 / 6

10 Fishburn numbers The Fishburn numbers ξ(n) have the following generating function. THM Zagier 200 n ( ( q) j ) = F ( q) n 0 ξ(n)q n = n 0 j= F (q) is Kontsevich s strange function (more later) The primitive Fishburn numbers have generating function F ( +q ). 5 / 6

11 Fishburn numbers The Fishburn numbers ξ(n) have the following generating function. THM Zagier 200 n ( ( q) j ) = F ( q) n 0 ξ(n)q n = n 0 j= F (q) is Kontsevich s strange function (more later) The primitive Fishburn numbers have generating function F ( +q ). Garvan introduces the numbers ξ r,s (n) by n 0 ξ r,s (n)q n = ( q) s n 0 n ( ( q) rj ) = ( q) s F (( q) r ). j= ξ(n) = ξ,0 (n) ( ) n ξ,0 (n) count primitive Fishburn matrices For instance, ξ,3 (n) = ξ(n) 3ξ(n ) + 3ξ(n 2) ξ(n 3). 5 / 6

12 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 R 6 / 6

13 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) R 6 / 6

14 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) R standard representation R 6 / 6

15 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) R Fishburn matrix M M i,j = # of intervals [i, j] standard representation R 6 / 6

16 Partition congruences Ramanujan proved the striking congruences p(5m ) 0 (mod 5), p(7m 2) 0 (mod 7), p(m 5) 0 (mod ). 7 / 6

17 Partition congruences Ramanujan proved the striking congruences p(5m ) 0 (mod 5), p(7m 2) 0 (mod 7), p(m 5) 0 (mod ). Also conjectured generalizations with moduli powers of 5, 7,. p(5 λ m δ 5 (λ)) 0 (mod 5 λ ), p(7 λ m δ 7 (λ)) 0 (mod 7 λ ), p( λ m δ (λ)) 0 (mod λ ), where δ p (λ) /24 (mod p λ ). 7 / 6

18 Partition congruences Ramanujan proved the striking congruences p(5m ) 0 (mod 5), p(7m 2) 0 (mod 7), p(m 5) 0 (mod ). Also conjectured generalizations with moduli powers of 5, 7,. p(5 λ m δ 5 (λ)) 0 (mod 5 λ ), p(7 λ m δ 7 (λ)) 0 (mod 7 λ/2 + ), p( λ m δ (λ)) 0 (mod λ ), where δ p (λ) /24 (mod p λ ). Watson (938), Atkin (967) 7 / 6

19 Partition congruences Ramanujan proved the striking congruences p(5m ) 0 (mod 5), p(7m 2) 0 (mod 7), p(m 5) 0 (mod ). Also conjectured generalizations with moduli powers of 5, 7,. p(5 λ m δ 5 (λ)) 0 (mod 5 λ ), p(7 λ m δ 7 (λ)) 0 (mod 7 λ/2 + ), p( λ m δ (λ)) 0 (mod λ ), where δ p (λ) /24 (mod p λ ). Watson (938), Atkin (967) RK There appear to be additional congruences, such as p(7 2 m 2, 9, 6, 30) 0 (mod 7 2 ), p(5 3 m, 26, 5) 0 (mod 5 3 ). 7 / 6

20 The Andrews Sellers congruences THM Andrews, Sellers 204 Let p be a prime, and j Z >0 such that ( ) 24k = for k =, 2,..., j. (AS) p Then, for all m, ξ(pm j) 0 (mod p). EG ξ(5m ) ξ(5m 2) 0 (mod 5) ξ(7m ) 0 (mod 7) ξ(m ) ξ(m 2) ξ(m 3) 0 (mod ) 8 / 6

21 The Andrews Sellers congruences THM Andrews, Sellers 204 Let p be a prime, and j Z >0 such that ( ) 24k = for k =, 2,..., j. (AS) p Then, for all m, ξ(pm j) 0 (mod p). EG ξ(5m ) ξ(5m 2) 0 (mod 5) ξ(7m ) 0 (mod 7) ξ(m ) ξ(m 2) ξ(m 3) 0 (mod ) Garvan (204) proved that (AS) may be replaced with ( ) 24k for k =, 2,..., j, p and that analogous congruences hold for the generalizations ξ r,s (n). 8 / 6

22 Garvan s generalized congruences Andrews Sellers congruences ξ(5m ) ξ(5m 2) 0 (mod 5) ξ(7m ) 0 (mod 7) Garvan s additional congruences ξ(23m ) ξ(23m 2)... ξ(23m 5) 0 (mod 23) Congruences for primitive Fishburn numbers Extensions observed by Garthwaite Rhoades ξ (5m ) 0 (mod 5) ξ(5m + 2) 2ξ(5m + ) 0 (mod 5) Garvan s congruences for r-fishburn numbers ξ 23 (7m ) ξ 23 (7m 2) ξ 23 (7m 3) 0 (mod 7) ξ (5m ) 2ξ (5m 2) + ξ (5m 3) 0 (mod 5) 9 / 6

23 Garvan s generalized congruences Andrews Sellers congruences ξ(5 λ m ) ξ(5 λ m 2) 0 (mod 5 λ ) ξ(7 λ m ) 0 (mod 7 λ ) Garvan s additional congruences ξ(23 λ m ) ξ(23 λ m 2)... ξ(23 λ m 5) 0 (mod 23 λ ) Congruences for primitive Fishburn numbers ξ (5m ) 0 (mod 5) x Extensions observed by Garthwaite Rhoades ξ(5 λ m + 2) 2ξ(5 λ m + ) 0 (mod 5 λ ) Garvan s congruences for r-fishburn numbers ξ 23 (7 λ m ) ξ 23 (7 λ m 2) ξ 23 (7 λ m 3) 0 (mod 7 λ ) ξ (5 λ m ) 2ξ (5 λ m 2) + ξ (5 λ m 3) 0 (mod 5 λ ) x Most, but not all, of these congruences can be lifted to prime powers. 9 / 6

24 Congruences modulo prime powers THM S 204 Let p be a prime, p r, and j Z >0 such that ( 24(k + s)/r Then, for all m and λ, p ) = for k =, 2,..., j. (C) ξ r,s (p λ m j) 0 (mod p λ ). 0 / 6

25 Congruences modulo prime powers THM S 204 Let p be a prime, p r, and j Z >0 such that ( 24(k + s)/r Then, for all m and λ, p ) = for k =, 2,..., j. (C) ξ r,s (p λ m j) 0 (mod p λ ). In (C), = may be replaced with, if p 5 and digit (s r/24; p) p. (*) (*) states that n p in s r/24 = n 0 + n p + n 2 p / 6

26 Congruences modulo prime powers THM S 204 Let p be a prime, p r, and j Z >0 such that ( 24(k + s)/r Then, for all m and λ, p ) = for k =, 2,..., j. (C) ξ r,s (p λ m j) 0 (mod p λ ). In (C), = may be replaced with, if p 5 and digit (s r/24; p) p. (*) EG Andrews Sellers conjectured and Garvan proved that ξ (5m ) 0 (mod 5). But (*) is not satisfied: s r/24 = /24 (mod 5 2 ) 0 / 6

27 Congruences modulo prime powers THM S 204 Let p be a prime, p r, and j Z >0 such that ( 24(k + s)/r Then, for all m and λ, p ) = for k =, 2,..., j. (C) ξ r,s (p λ m j) 0 (mod p λ ). In (C), = may be replaced with, if p 5 and digit (s r/24; p) p. (*) EG Andrews Sellers conjectured and Garvan proved that ξ (5m ) 0 (mod 5). But (*) is not satisfied: s r/24 = /24 (mod 5 2 ) Indeed, the congruences do not extend to prime powers: ξ (5 2 ) = (mod 5 2 ) 0 / 6

28 Primitive Fishburn matrices Q Can we give a combinatorial interpretation for any of these congruences? In particular, for the number of primitive Fishburn matrices, ξ (5m ) 0 (mod 5). / 6

29 Primitive Fishburn matrices Q Can we give a combinatorial interpretation for any of these congruences? In particular, for the number of primitive Fishburn matrices, ξ (5m ) 0 (mod 5). The Ramanujan congruences (Atkin, Swinnerton-Dyer (954)) p(5m ) 0 (mod 5), p(7m 2) 0 (mod 7), p(m 5) 0 (mod ). modulo 5 and 7 are explained by Dyson s rank. (rank = largest part of a partition minus the number of its parts) / 6

30 Primitive Fishburn matrices Q Can we give a combinatorial interpretation for any of these congruences? In particular, for the number of primitive Fishburn matrices, ξ (5m ) 0 (mod 5). The Ramanujan congruences (Atkin, Swinnerton-Dyer (954)) p(5m ) 0 (mod 5), p(7m 2) 0 (mod 7), p(m 5) 0 (mod ). modulo 5 and 7 are explained by Dyson s rank. (rank = largest part of a partition minus the number of its parts) All three congruences are explained by Dyson s speculated crank, which was found by Andrews and Garvan (988). / 6

31 Kontsevich s strange function DEF F (q) = n 0( q)( q 2 ) ( q n ) does not converge in any open set series terminates when q is a root of unity 2 / 6

32 Kontsevich s strange function DEF F (q) = n 0( q)( q 2 ) ( q n ) does not converge in any open set series terminates when q is a root of unity THM Zagier 999 The following strange identity holds : q /24 F (q) = 2 ( ) 2 n q n2 /24 n n= LHS agrees at roots of unity with the radial limit of the RHS (and similarly for the derivatives of all orders) 2 / 6

33 Kontsevich s strange function DEF F (q) = n 0( q)( q 2 ) ( q n ) does not converge in any open set series terminates when q is a root of unity THM Zagier 999 The following strange identity holds : q /24 F (q) = 2 ( ) 2 n q n2 /24 n n= LHS agrees at roots of unity with the radial limit of the RHS (and similarly for the derivatives of all orders) RHS is the half-derivative (up to constants) of the Dedekind eta function η(τ) = q ( ) 2 /24 q n ( n ) = q n2 /24. n n= 2 / 6

34 Eichler integrals of half-integral weight If f(τ) = a(n)q n is a cusp form of integral weight k on SL 2 (Z), then f(τ) = n k a(n)q n is its Eichler integral. f(τ + ) = f(τ), τ k 2 f( /τ) = f(τ) + poly(τ). 3 / 6

35 Eichler integrals of half-integral weight If f(τ) = a(n)q n is a cusp form of integral weight k on SL 2 (Z), then f(τ) = n k a(n)q n is its Eichler integral. f(τ + ) = f(τ), τ k 2 f( /τ) = f(τ) + poly(τ). EG Eisenstein series σ k (n)q n n= integrate n= σ k (n) n k q n = σ k (n)q n n= 3 / 6

36 Eichler integrals of half-integral weight If f(τ) = a(n)q n is a cusp form of integral weight k on SL 2 (Z), then f(τ) = n k a(n)q n is its Eichler integral. f(τ + ) = f(τ), τ k 2 f( /τ) = f(τ) + poly(τ). EG Eisenstein series σ k (n)q n n= integrate n= σ k (n) n k q n = σ k (n)q n n= η(τ) = ( 2 ) n q n 2 /24 has weight k = /2. Its formal Eichler integral is η(τ) = ( 24 n 2 ) n q n 2 /24 = 6 q /24 F (q). 3 / 6

37 Eichler integrals of half-integral weight If f(τ) = a(n)q n is a cusp form of integral weight k on SL 2 (Z), then f(τ) = n k a(n)q n is its Eichler integral. f(τ + ) = f(τ), τ k 2 f( /τ) = f(τ) + poly(τ). EG Eisenstein series σ k (n)q n n= integrate n= σ k (n) n k q n = σ k (n)q n n= η(τ) = ( 2 ) n q n 2 /24 has weight k = /2. Its formal Eichler integral is η(τ) = ( 24 n 2 ) n q n 2 /24 = 6 q /24 F (q). Bringmann Rolen (204) study Eichler integrals of half-integral weight systematically. (inspired by examples of Lawrence Zagier, 999) 3 / 6

38 Quantum modular forms THM Bringmann Rolen 204 If f(τ) is a formal Eichler integral of half-integral weight, then f(τ) is a quantum modular form. 4 / 6

39 Quantum modular forms THM Bringmann Rolen 204 If f(τ) is a formal Eichler integral of half-integral weight, then f(τ) is a quantum modular form. According to an intentionally vague definition of Zagier (200), a quantum modular form is a function f : P (Q) C with f(τ) (cτ+d) k f( aτ+b cτ+d ) = nice(τ). 4 / 6

40 Quantum modular forms THM Bringmann Rolen 204 If f(τ) is a formal Eichler integral of half-integral weight, then f(τ) is a quantum modular form. According to an intentionally vague definition of Zagier (200), a quantum modular form is a function f : P (Q) C with f(τ) (cτ+d) k f( aτ+b cτ+d ) = nice(τ). Above, nice(τ) is defined and real analytic on R (except at one point). 4 / 6

41 Congruences for coefficients of quantum modular forms Is there a general theory of congruences for the coefficients A n of f(τ) = A n ( q) n if f is a quantum modular form? In particular, if f is the Eichler integral of a half-integral weight modular form? What about expansions f(τ) = B n (ζ q) n at other roots of unity? 5 / 6

42 Congruences for coefficients of quantum modular forms Is there a general theory of congruences for the coefficients A n of f(τ) = A n ( q) n if f is a quantum modular form? In particular, if f is the Eichler integral of a half-integral weight modular form? What about expansions f(τ) = B n (ζ q) n at other roots of unity? Inspiring results by Guerzhoy Kent Rolen (204) for Eichler integrals of certain unary theta series. 5 / 6

43 Congruences for coefficients of quantum modular forms Is there a general theory of congruences for the coefficients A n of f(τ) = A n ( q) n if f is a quantum modular form? In particular, if f is the Eichler integral of a half-integral weight modular form? What about expansions f(τ) = B n (ζ q) n at other roots of unity? Inspiring results by Guerzhoy Kent Rolen (204) for Eichler integrals of certain unary theta series. In all cases, is it true that the known congruences are complete? 5 / 6

44 THANK YOU! Slides for this talk will be available from my website: S. Ahlgren, B. Kim Dissections of a strange function Preprint, 204 G. E. Andrews, J. A. Sellers Congruences for the Fishburn numbers Preprint, 204 F. G. Garvan Congruences and relations for r-fishburn numbers Preprint, 204 P. Guerzhoy, Z. A. Kent, L. Rolen Congruences for Taylor expansions of quantum modular forms Preprint, 204 A. Straub Preprint, / 6

45 Ingredients of proof Crucial ingredients: N p F (q) (q; q) n = q i A p (N, i, q p ) n=0 i=0 7 / 8

46 Ingredients of proof N p Crucial ingredients: F (q) (q; q) n = q i A p (N, i, q p ) If i S(p) and i 0 /24 modulo p, then n=0 i=0 A p (pn, i, q) = ( q) n α p (n, i, q), ( ) A p (pn, i 0, q) = 2 p pq p/24 F (q p, pn ) + ( q) n β p (n, q). (Andrews Sellers) (Garvan) where α p Z[q] and β p Z[q]. 7 / 8

47 Ingredients of proof N p Crucial ingredients: F (q) (q; q) n = q i A p (N, i, q p ) If i S(p) and i 0 /24 modulo p, then n=0 i=0 A p (pn, i, q) = ( q) n α p (n, i, q), ( ) A p (pn, i 0, q) = 2 p pq p/24 F (q p, pn ) + ( q) n β p (n, q). (Andrews Sellers) (Garvan) where α p Z[q] and β p Z[q]. Ahlgren Kim (204) show that, in fact, (q; q) n divides A p (pn, i, q). 7 / 8

48 Partition congruences Atkin (968) proved further congruences for small prime moduli. EG p(3 3 m + 237) 0 (mod 3) p( m ) 0 (mod 23) 8 / 8

49 Partition congruences Atkin (968) proved further congruences for small prime moduli. EG p(3 3 m + 237) 0 (mod 3) p( m ) 0 (mod 23) Ono (2000) and Ahlgren Ono (200) show that, if M is coprime to 6, then p(am + B) 0 (mod M) for infinitely many non-nested arithmetic progressions Am + B. It is conjectured that no congruences exist for moduli 2 and 3. 8 / 8

Congruences for Fishburn numbers modulo prime powers

Congruences for Fishburn numbers modulo prime powers Armin Straub Department of Mathematics University of Illinois at Urbana-Champaign July 16, 2014 Abstract The Fishburn numbers ξ(n) are defined by the