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Arithmetic of k-regular partition
functions
David Penniston, UW Oshkosh
Arithmetic of k-regular partition functions – p. 1
A partition of n is a way of writing n as a sum ofpositive integers, where the ordering of the integers isirrelevant
Arithmetic of k-regular partition functions – p. 2
Partitions of 4
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
Arithmetic of k-regular partition functions – p. 3
Partition function
p(n) := number of partitions of n
Arithmetic of k-regular partition functions – p. 4
Partition function
p(n) := number of partitions of n
p(4) = 5
Arithmetic of k-regular partition functions – p. 4
n p(n)
10 42
20 627
30 5604
40 37338
50 204226
Arithmetic of k-regular partition functions – p. 5
The arithmetic of p(n)
Arithmetic of k-regular partition functions – p. 6
The arithmetic of p(n)
p(0) p(1) p(2) p(3) p(4)
p(5) p(6) p(7) p(8) p(9)
p(10) p(11) p(12) p(13) p(14)
p(15) p(16) p(17) p(18) p(19)
p(20) p(21) p(22) p(23) p(24)
p(25) p(26) p(27) p(28) p(29)
p(30) p(31) p(32) p(33) p(34)
p(35) p(36) p(37) p(38) p(39)
p(40) p(41) p(42) p(43) p(44)
p(45) p(46) p(47) p(48) p(49)
Arithmetic of k-regular partition functions – p. 6
The arithmetic of p(n)
1 1 2 3 5
7 11 15 22 30
42 56 77 101 135
176 231 297 385 490
627 792 1002 1255 1575
1958 2436 3010 3718 4565
5604 6842 8349 10143 12310
14883 17977 21637 26015 31185
37338 44583 53174 63261 75175
89134 105558 124754 147273 173525
Arithmetic of k-regular partition functions – p. 7
1 1 2 3 5
7 11 15 22 30
42 56 77 101 135
176 231 297 385 490
627 792 1002 1255 1575
1958 2436 3010 3718 4565
5604 6842 8349 10143 12310
14883 17977 21637 26015 31185
37338 44583 53174 63261 75175
89134 105558 124754 147273 173525
Arithmetic of k-regular partition functions – p. 8
p(0) p(1) p(2) p(3) p(4)
p(5) p(6) p(7) p(8) p(9)
p(10) p(11) p(12) p(13) p(14)
p(15) p(16) p(17) p(18) p(19)
p(20) p(21) p(22) p(23) p(24)
p(25) p(26) p(27) p(28) p(29)
p(30) p(31) p(32) p(33) p(34)
p(35) p(36) p(37) p(38) p(39)
p(40) p(41) p(42) p(43) p(44)
p(45) p(46) p(47) p(48) p(49)
Arithmetic of k-regular partition functions – p. 9
1st Ramanujan congruence
For every n ≥ 0,
p(5n+ 4) is divisible by 5
Arithmetic of k-regular partition functions – p. 10
Ramanujan congruences
p(5n+ 4) is divisible by 5
p(7n+ 5) is divisible by 7
p(11n+ 6) is divisible by 11
Arithmetic of k-regular partition functions – p. 11
Other congruences?
Arithmetic of k-regular partition functions – p. 12
Other congruences?
(Atkin-O’Brien)
p(157525693n + 111247) is divisible by 13
Arithmetic of k-regular partition functions – p. 12
Other congruences?
(Atkin-O’Brien)
p(157525693n + 111247) is divisible by 13
(p(111247) is a number with well over 300 digits)
Arithmetic of k-regular partition functions – p. 12
(K. Ono)
For every prime m ≥ 5, there exist positive integers Aand B such that
p(An+ B) is divisible by m
Arithmetic of k-regular partition functions – p. 13
What about 2 and 3?
Arithmetic of k-regular partition functions – p. 14
Of the first 106 values of p(n),
Arithmetic of k-regular partition functions – p. 15
Of the first 106 values of p(n),
50.0446% are divisible by 2
Arithmetic of k-regular partition functions – p. 15
Of the first 106 values of p(n),
50.0446% are divisible by 2
33.3012% are divisible by 3
Arithmetic of k-regular partition functions – p. 15
A partition is called k-regular if none of its parts isdivisible by k
Arithmetic of k-regular partition functions – p. 16
A partition is called k-regular if none of its parts isdivisible by k
bk(n) := number of k-regular partitions of n
Arithmetic of k-regular partition functions – p. 16
p(4) = 5
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
Arithmetic of k-regular partition functions – p. 17
b2(4) = 2
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
Arithmetic of k-regular partition functions – p. 18
b3(4) = 4
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
Arithmetic of k-regular partition functions – p. 19
b2(n)
1 1 1 2 2
3 4 5 6 8
10 12 15 18 22
27 32 38 46 54
64 76 89 104 122
142 165 192 222 256
296 340 390 448 512
585 668 760 864 982
1113 1260 1426 1610 1816
2048 2304 2590 2910 3264
Arithmetic of k-regular partition functions – p. 20
b2(n)
1 1 1 2 2
3 4 5 6 8
10 12 15 18 22
27 32 38 46 54
64 76 89 104 122
142 165 192 222 256
296 340 390 448 512
585 668 760 864 982
1113 1260 1426 1610 1816
2048 2304 2590 2910 3264
Arithmetic of k-regular partition functions – p. 21
n
0 1 2 3 4
5 6 7 8 9
10 11 12 13 14
15 16 17 18 19
20 21 22 23 24
25 26 27 28 29
30 31 32 33 34
35 36 37 38 39
40 41 42 43 44
45 46 47 48 49
Arithmetic of k-regular partition functions – p. 22
b2(n) is odd
⇐⇒
n ∈ {0, 1, 2, 5, 7, 12, 15, 22, 26, . . .}
Arithmetic of k-regular partition functions – p. 23
b2(n) is odd
⇐⇒
n ∈ {0} ∪ {1, 2} ∪ {5, 7} ∪ {12, 15} ∪ · · ·
Arithmetic of k-regular partition functions – p. 24
b2(n) is odd
⇐⇒
n ∈ {0} ∪ {1, 2} ∪ {5, 7} ∪ {12, 15} ∪ · · ·
⇐⇒
n =ℓ(3ℓ+ 1)
2(ℓ ∈ Z)
Arithmetic of k-regular partition functions – p. 24
b13(n)
1 1 2 3 5 7
11 15 22 30 42 56
77 100 134 174 228 292
378 479 612 770 972 1213
1519 1881 2334 2874 3540 4331
5302 6450 7848 9501 11496 13851
16680 20006 23980 28648 34193 40689
48378 57360 67948 80295 94788 111652
Arithmetic of k-regular partition functions – p. 25
b13(n) is odd
⇐⇒
n ∈ {0, 1, 3, 4, 5, 6, 7, 12, 19, 23, 24, 25, 29, . . .}
Arithmetic of k-regular partition functions – p. 26
b13(2n) is odd
⇐⇒
2n ∈ {0, 4, 6, 12, 24, 40, 60, 84, 112, . . .}
Arithmetic of k-regular partition functions – p. 27
b13(2n) is odd
⇐⇒
2n ∈ {0, 4, 12, 24, 40, 60, 84, 112, . . .}
or 2n ∈ {6, 58, 162, 318, 526, 786, . . .}
Arithmetic of k-regular partition functions – p. 28
b13(2n) is odd
⇐⇒
n/2 ∈ {0, 1, 3, 6, 10, 15, 21, 28, . . .}
or 2n ∈ {6, 58, 162, 318, 526, 786, . . .}
Arithmetic of k-regular partition functions – p. 29
b13(2n) is odd
⇐⇒
n/2 =ℓ(ℓ+ 1)
2(ℓ ∈ N)
or 2n ∈ {6, 58, 162, 318, 526, 786, . . .}
Arithmetic of k-regular partition functions – p. 30
b13(2n) is odd
⇐⇒
n = ℓ(ℓ+ 1) (ℓ ∈ N)
or 2n ∈ {6, 58, 162, 318, 526, 786, . . .}
Arithmetic of k-regular partition functions – p. 31
b13(2n) is odd
⇐⇒
n = ℓ(ℓ+ 1) (ℓ ∈ N)
or 2n− 6 ∈ {0, 52, 156, 312, 520, 780, . . .}
Arithmetic of k-regular partition functions – p. 32
b13(2n) is odd
⇐⇒
n = ℓ(ℓ+ 1) (ℓ ∈ N)
or 2n−652 ∈ {0, 1, 3, 6, 10, 15, . . .}
Arithmetic of k-regular partition functions – p. 33
b13(2n) is odd
⇐⇒
n = ℓ(ℓ+ 1) (ℓ ∈ N)
or 2n−652 = ℓ(ℓ+1)
2
Arithmetic of k-regular partition functions – p. 34
(Calkin, Drake, James, Law, Lee, P., Radder)
b13(2n) is odd
⇐⇒
n = ℓ(ℓ+ 1) (ℓ ∈ N)
or n = 13ℓ(ℓ+ 1) + 3 (ℓ ∈ N)
Arithmetic of k-regular partition functions – p. 35
b13(n)
1 1 2 3 5 7
11 15 22 30 42 56
77 100 134 174 228 292
378 479 612 770 972 1213
1519 1881 2334 2874 3540 4331
5302 6450 7848 9501 11496 13851
16680 20006 23980 28648 34193 40689
48378 57360 67948 80295 94788 111652
Arithmetic of k-regular partition functions – p. 36
b13(n) mod 3
1 1 2 0 2 1
2 0 1 0 0 2
2 1 2 0 0 1
0 2 0 2 0 1
1 0 0 0 0 2
1 0 0 0 0 0
0 2 1 1 2 0
0 0 1 0 0 1
Arithmetic of k-regular partition functions – p. 37
b13(n) mod 3
1 1 2 0 2 1 2 0 1
0 0 2 2 1 2 0 0 1
0 2 0 2 0 1 1 0 0
0 0 2 1 0 0 0 0 0
0 2 1 1 2 0 0 0 1
0 0 1 0 0 0 2 0 2
0 0 2 0 1 1 2 0 1
1 2 0 0 0 0 2 0 2
0 1 2 0 0 2 2 0 2
0 0 0 1 0 1 0 0 2
2 1 0 0 0 0 0 0 1
Arithmetic of k-regular partition functions – p. 38
b13(3n + 1) mod 3
1 1 2 3 2 1 2 0 1
0 0 2 2 1 2 0 0 1
0 2 0 2 0 1 1 0 0
0 0 2 1 0 0 0 0 0
0 2 1 1 2 0 0 0 1
0 0 1 0 0 0 2 0 2
0 0 2 0 1 1 2 0 1
1 2 0 0 0 0 2 0 2
0 1 2 0 0 2 2 0 2
0 0 0 1 0 1 0 0 2
2 1 0 0 0 0 0 0 1
Arithmetic of k-regular partition functions – p. 39
b13(3n+ 1) (mod 3)
1, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, . . .
Arithmetic of k-regular partition functions – p. 40
b13(3n + 1) mod 3
1 1 2 3 2 1 2 0 1
0 0 2 2 1 2 0 0 1
0 2 0 2 0 1 1 0 0
0 0 2 1 0 0 0 0 0
0 2 1 1 2 0 0 0 1
0 0 1 0 0 0 2 0 2
0 0 2 0 1 1 2 0 1
1 2 0 0 0 0 2 0 2
0 1 2 0 0 2 2 0 2
0 0 0 1 0 1 0 0 2
2 1 0 0 0 0 0 0 1
Arithmetic of k-regular partition functions – p. 41
b13(9n + 4) mod 3
1 1 2 3 2 1 2 0 1
0 0 2 2 1 2 0 0 1
0 2 0 2 0 1 1 0 0
0 0 2 1 0 0 0 0 0
0 2 1 1 2 0 0 0 1
0 0 1 0 0 0 2 0 2
0 0 2 0 1 1 2 0 1
1 2 0 0 0 0 2 0 2
0 1 2 0 0 2 2 0 2
0 0 0 1 0 1 0 0 2
2 1 0 0 0 0 0 0 1
Arithmetic of k-regular partition functions – p. 42
b13(3n+ 1) (mod 3)
1, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, . . .
b13(9n+ 4) (mod 3)
2, 1, 0, 0, 2, 0, 1, 0, 0, 0, 0, . . .
Arithmetic of k-regular partition functions – p. 43
b13(9n+ 4) + b13(3n+ 1) ≡ 0 (mod 3)
Arithmetic of k-regular partition functions – p. 44
b13(3n + 1) mod 3
1 1 2 3 2 1 2 0 1
0 0 2 2 1 2 0 0 1
0 2 0 2 0 1 1 0 0
0 0 2 1 0 0 0 0 0
0 2 1 1 2 0 0 0 1
0 0 1 0 0 0 2 0 2
0 0 2 0 1 1 2 0 1
1 2 0 0 0 0 2 0 2
0 1 2 0 0 2 2 0 2
0 0 0 1 0 1 0 0 2
2 1 0 0 0 0 0 0 1
Arithmetic of k-regular partition functions – p. 45
b13(9n + 7) mod 3
1 1 2 3 2 1 2 0 1
0 0 2 2 1 2 0 0 1
0 2 0 2 0 1 1 0 0
0 0 2 1 0 0 0 0 0
0 2 1 1 2 0 0 0 1
0 0 1 0 0 0 2 0 2
0 0 2 0 1 1 2 0 1
1 2 0 0 0 0 2 0 2
0 1 2 0 0 2 2 0 2
0 0 0 1 0 1 0 0 2
2 1 0 0 0 0 0 0 1
Arithmetic of k-regular partition functions – p. 46
b13(9n+ 4) + b13(3n+ 1) ≡ 0 (mod 3)
b13(9ℓ+ 7) ≡ 0 (mod 3)
Arithmetic of k-regular partition functions – p. 47
(Calkin, Drake, James, Law, Lee, P., Radder)
For every 2 ≤ s ≤ 6,
b13
(
3sn+
(
5 · 3s−1 − 1
2
))
≡ 0 (mod 3)
Arithmetic of k-regular partition functions – p. 48
(Webb)
For every s ≥ 2,
b13
(
3sn+
(
5 · 3s−1 − 1
2
))
≡ 0 (mod 3)
Arithmetic of k-regular partition functions – p. 49
(Andrews, Hirschhorn, Sellers)
For every s ≥ 1,
b4
(
32sn+
(
19 · 32s−1 − 1
8
))
≡ 0 (mod 3)
Arithmetic of k-regular partition functions – p. 50
(Furcy, P.)
For each k ∈ {7, 19, 25, 34, 37, 43, 49}, there exists an
analogous family of congruences for bk(n) modulo 3.
Arithmetic of k-regular partition functions – p. 51
(Furcy, P.)
For each k ∈ {7, 19, 25, 34, 37, 43, 49}, there exists an
analogous family of congruences for bk(n) modulo 3.
For example,
b25(32s+1n+ (2 · 32s − 1)) ≡ 0 (mod 3)
Arithmetic of k-regular partition functions – p. 51
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