A partition of a positive integer n is any way of writing as a sum of one or more positive integers, in which we don't care about the order of the numbers in the sum.

For example, the number 4 can be written as a sum of one or more positive integers (where we don't care about the order of the numbers in the sum) in exactly five ways:

\[4,\; 3 + 1,\; 2 + 2,\; 2 + 1 + 1,\; 1 + 1 + 1 + 1.\]

So 4 has five partitions.

What is the number of partitions of the number 7?

Darkside Feb 4, 2019

#1**0 **

**Partitions of (7) =15 as follows: 7 = 7 6 + 1 = 7 5 + 2 = 7 5 + 1 + 1 = 7 4 + 3 = 7 4 + 2 + 1 = 7 4 + 1 + 1 + 1 = 7 3 + 3 + 1 = 7 3 + 2 + 2 = 7 3 + 2 + 1 + 1 = 7 3 + 1 + 1 + 1 + 1 = 7 2 + 2 + 2 + 1 = 7 2 + 2 + 1 + 1 + 1 = 7 2 + 1 + 1 + 1 + 1 + 1 = 7 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7**

Guest Feb 4, 2019

#2**+1 **

I ran this Python code with GCC 4.8.2 on Linux:

from sympy.ntheory import npartitions

for i in range(1,100): print(npartitions(i))

Output:

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

204226

239943

281589

329931

386155

451276

526823

614154

715220

831820

966467

1121505

1300156

1505499

1741630

2012558

2323520

2679689

3087735

3554345

4087968

4697205

5392783

6185689

7089500

8118264

9289091

10619863

12132164

13848650

15796476

18004327

20506255

23338469

26543660

30167357

34262962

38887673

44108109

49995925

56634173

64112359

72533807

82010177

92669720

104651419

118114304

133230930

150198136

169229875

The 7th row is 15. So the answer is 15 :P

MaxWong Feb 7, 2019