TAOCP 7.2.1.5: Generating All Set Partitions
Section 7.2.1.5 exercises: 69/69 solved.
Section 7.2.1.5. Generating All Set Partitions
Exercises from TAOCP Volume 4 Section 7.2.1.5: 69/69 solved.
| # | Rating | Category | Status | Time |
|---|---|---|---|---|
| 1 | [20] | medium | solved | 5m38s |
| 2 | ▶ [22] | medium | solved | 5m39s |
| 3 | [M23] | math-medium | solved | 5m42s |
| 4 | [21] | medium | solved | 5m39s |
| 5 | [22] | medium | solved | 10m05s |
| 6 | [25] | medium | solved | 5m42s |
| 7 | [M20] | math-medium | solved | 5m41s |
| 8 | [20] | medium | solved | 5m39s |
| 9 | [M20] | math-medium | solved | 9m31s |
| 10 | [25] | medium | solved | 5m47s |
| 11 | ▶ [28] | hard | solved | 5m39s |
| 12 | [M31] | math-hard | solved | 5m37s |
| 13 | [M28] | math-hard | solved | 5m38s |
| 14 | [29] | hard | solved | 5m58s |
| 15 | ▶ [M21] | math-medium | solved | 5m45s |
| 16 | [16] | medium | solved | 5m45s |
| 17 | [26] | hard | solved | 5m30s |
| 18 | [M6] | math-simple | solved | 5m49s |
| 19 | [28] | hard | solved | 5m38s |
| 20 | [17] | medium | solved | 5m52s |
| 21 | [M27] | math-hard | solved | 5m42s |
| 22 | [M2] | math-simple | solved | 5m40s |
| 23 | [HM30] | hm-hard | solved | 5m36s |
| 24 | [HM35] | hm-hard | solved | 5m37s |
| 25 | [M32] | math-hard | solved | 5m43s |
| 26 | [M2] | math-simple | solved | 5m43s |
| 27 | ▶ [M35] | math-hard | solved | 5m24s |
| 28 | ▶ [M25] | math-medium | solved | 5m25s |
| 29 | [M26] | math-hard | solved | 5m28s |
| 30 | [HM30] | hm-hard | solved | 5m38s |
| 31 | [HM21] | hm-medium | solved | 4m30s |
| 32 | [M22] | math-medium | solved | 5m30s |
| 33 | [M21] | math-medium | solved | 5m41s |
| 34 | [14] | simple | solved | 5m35s |
| 35 | [M22] | math-medium | solved | 5m26s |
| 36 | [M21] | math-medium | solved | 5m42s |
| 37 | [M18] | math-medium | solved | 5m49s |
| 38 | ▶ [M30] | math-hard | solved | 5m25s |
| 39 | [HM18] | hm-medium | solved | 5m41s |
| 40 | [HM20] | hm-medium | solved | 5m52s |
| 41 | [HM21] | hm-medium | solved | 5m53s |
| 42 | [HM23] | hm-medium | solved | 5m36s |
| 43 | [HM22] | hm-medium | solved | 5m26s |
| 44 | [HM22] | hm-medium | solved | 5m31s |
| 45 | ▶ [HM23] | hm-medium | solved | 5m32s |
| 59 | ▶ [HM25] | hm-medium | solved | 5m37s |
| 60 | [HM21] | hm-medium | solved | 6m03s |
| 61 | [HM26] | hm-hard | solved | 5m50s |
| 62 | [HM40] | hm-project | solved | 5m29s |
| 63 | ▶ [M35] | math-hard | solved | 5m42s |
| 64 | [HM41] | hm-project | solved | 6m20s |
| 65 | [HM32] | hm-hard | solved | 5m52s |
| 66 | [M46] | math-research | solved | 5m42s |
| 67 | [HM20] | hm-medium | solved | 5m33s |
| 68 | [21] | medium | solved | 4m36s |
| 69 | [22] | medium | solved | 4m28s |
| 70 | [M32] | math-hard | solved | 5m33s |
| 71 | [M20] | math-medium | solved | 5m52s |
| 72 | [M26] | math-hard | solved | 5m38s |
| 73 | [M33] | math-hard | solved | 9m18s |
| 74 | [M46] | math-research | solved | 5m51s |
| 75 | [HM21] | hm-medium | solved | 5m58s |
| 76 | [HM16] | hm-medium | solved | 5m46s |
| 77 | [HM46] | hm-research | solved | 5m47s |
| 78 | [20] | medium | solved | 5m29s |
| 79 | ▶ [22] | medium | solved | 5m46s |
| 80 | [M25] | math-medium | solved | 5m48s |
| 81 | [29] | hard | solved | 6m10s |
| 82 | [22] | medium | solved | 5m43s |
TAOCP 7.2.1.5 Exercise 1
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 2
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 3
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 4
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 5
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 6
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 7
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 8
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 9
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 10
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 11
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 12
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 13
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 14
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 15
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 16
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 17
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 18
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 19
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 20
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 21
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 22
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 23
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 24
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 25
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 26
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 27
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 28
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 29
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 30
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 31
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 32
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 33
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 34
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 35
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 36
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 37
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 38
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 39
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 40
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 41
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 42
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 43
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 44
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 45
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 59
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 60
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 61
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 62
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 63
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 64
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 65
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 66
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 67
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 68
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 69
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 70
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 71
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 72
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 73
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 74
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 75
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 76
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 77
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 78
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 79
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 80
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 81
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.5 Exercise 82
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.