Background
Given any $n \in \mathbb{N}$, the ordered factorization count of $n$ can be computed and is traditionally written $H(n)$. This is, essentially, the number of unique decompositions of $n$ into products of not-necessarily-prime factors.
For example, the number 30 can be decomposed 13 different ways: { 2×3×5, 2×5×3, 2×15, 3×2×5, 3×5×2, 3×10, 5×2×3, 5×3×2, 5×6, 6×5, 10×3, 15×2, 30 }.
Question
Let $F(n,i)$ be the $i$th ordered factorization of $n$, where $1 \le i \le H(n)$. When enumerating the set of all ordered factorizations of $n$, what notation is used to codify the tuple $(n,i)$, that is, how do we denote “the $i$th factorization of $n$” succinctly?
I have been using the invented syntax n#i. But what is the standard notation in mathematics for this concept?
Here is another example:
120#1 = 2 x 2 x 2 x 3 x 5
120#2 = 2 x 2 x 2 x 5 x 3
120#3 = 2 x 2 x 2 x 15
120#4 = 2 x 2 x 3 x 2 x 5
120#5 = 2 x 2 x 3 x 5 x 2
120#6 = 2 x 2 x 3 x 10
120#7 = 2 x 2 x 5 x 2 x 3
120#8 = 2 x 2 x 5 x 3 x 2
120#9 = 2 x 2 x 5 x 6
120#10 = 2 x 2 x 6 x 5
120#11 = 2 x 2 x 10 x 3
120#12 = 2 x 2 x 15 x 2
120#13 = 2 x 2 x 30
120#14 = 2 x 3 x 2 x 2 x 5
120#15 = 2 x 3 x 2 x 5 x 2
120#16 = 2 x 3 x 2 x 10
120#17 = 2 x 3 x 4 x 5
120#18 = 2 x 3 x 5 x 2 x 2
120#19 = 2 x 3 x 5 x 4
120#20 = 2 x 3 x 10 x 2
120#21 = 2 x 3 x 20
120#22 = 2 x 4 x 3 x 5
120#23 = 2 x 4 x 5 x 3
120#24 = 2 x 4 x 15
120#25 = 2 x 5 x 2 x 2 x 3
120#26 = 2 x 5 x 2 x 3 x 2
120#27 = 2 x 5 x 2 x 6
120#28 = 2 x 5 x 3 x 2 x 2
120#29 = 2 x 5 x 3 x 4
120#30 = 2 x 5 x 4 x 3
120#31 = 2 x 5 x 6 x 2
120#32 = 2 x 5 x 12
120#33 = 2 x 6 x 2 x 5
120#34 = 2 x 6 x 5 x 2
120#35 = 2 x 6 x 10
120#36 = 2 x 10 x 2 x 3
120#37 = 2 x 10 x 3 x 2
120#38 = 2 x 10 x 6
120#39 = 2 x 12 x 5
120#40 = 2 x 15 x 2 x 2
120#41 = 2 x 15 x 4
120#42 = 2 x 20 x 3
120#43 = 2 x 30 x 2
120#44 = 2 x 60
120#45 = 3 x 2 x 2 x 2 x 5
120#46 = 3 x 2 x 2 x 5 x 2
120#47 = 3 x 2 x 2 x 10
120#48 = 3 x 2 x 4 x 5
120#49 = 3 x 2 x 5 x 2 x 2
120#50 = 3 x 2 x 5 x 4
120#51 = 3 x 2 x 10 x 2
120#52 = 3 x 2 x 20
120#53 = 3 x 4 x 2 x 5
120#54 = 3 x 4 x 5 x 2
120#55 = 3 x 4 x 10
120#56 = 3 x 5 x 2 x 2 x 2
120#57 = 3 x 5 x 2 x 4
120#58 = 3 x 5 x 4 x 2
120#59 = 3 x 5 x 8
120#60 = 3 x 8 x 5
120#61 = 3 x 10 x 2 x 2
120#62 = 3 x 10 x 4
120#63 = 3 x 20 x 2
120#64 = 3 x 40
120#65 = 4 x 2 x 3 x 5
120#66 = 4 x 2 x 5 x 3
120#67 = 4 x 2 x 15
120#68 = 4 x 3 x 2 x 5
120#69 = 4 x 3 x 5 x 2
120#70 = 4 x 3 x 10
120#71 = 4 x 5 x 2 x 3
120#72 = 4 x 5 x 3 x 2
120#73 = 4 x 5 x 6
120#74 = 4 x 6 x 5
120#75 = 4 x 10 x 3
120#76 = 4 x 15 x 2
120#77 = 4 x 30
120#78 = 5 x 2 x 2 x 2 x 3
120#79 = 5 x 2 x 2 x 3 x 2
120#80 = 5 x 2 x 2 x 6
120#81 = 5 x 2 x 3 x 2 x 2
120#82 = 5 x 2 x 3 x 4
120#83 = 5 x 2 x 4 x 3
120#84 = 5 x 2 x 6 x 2
120#85 = 5 x 2 x 12
120#86 = 5 x 3 x 2 x 2 x 2
120#87 = 5 x 3 x 2 x 4
120#88 = 5 x 3 x 4 x 2
120#89 = 5 x 3 x 8
120#90 = 5 x 4 x 2 x 3
120#91 = 5 x 4 x 3 x 2
120#92 = 5 x 4 x 6
120#93 = 5 x 6 x 2 x 2
120#94 = 5 x 6 x 4
120#95 = 5 x 8 x 3
120#96 = 5 x 12 x 2
120#97 = 5 x 24
120#98 = 6 x 2 x 2 x 5
120#99 = 6 x 2 x 5 x 2
120#100 = 6 x 2 x 10
120#101 = 6 x 4 x 5
120#102 = 6 x 5 x 2 x 2
120#103 = 6 x 5 x 4
120#104 = 6 x 10 x 2
120#105 = 6 x 20
120#106 = 8 x 3 x 5
120#107 = 8 x 5 x 3
120#108 = 8 x 15
120#109 = 10 x 2 x 2 x 3
120#110 = 10 x 2 x 3 x 2
120#111 = 10 x 2 x 6
120#112 = 10 x 3 x 2 x 2
120#113 = 10 x 3 x 4
120#114 = 10 x 4 x 3
120#115 = 10 x 6 x 2
120#116 = 10 x 12
120#117 = 12 x 2 x 5
120#118 = 12 x 5 x 2
120#119 = 12 x 10
120#120 = 15 x 2 x 2 x 2
120#121 = 15 x 2 x 4
120#122 = 15 x 4 x 2
120#123 = 15 x 8
120#124 = 20 x 2 x 3
120#125 = 20 x 3 x 2
120#126 = 20 x 6
120#127 = 24 x 5
120#128 = 30 x 2 x 2
120#129 = 30 x 4
120#130 = 40 x 3
120#131 = 60 x 2
120#132 = 120
Note that this question is similar to, but not a duplicate of, the related question: What is the pair $(n,k)$ called where $n$ is and integer and $k$ is the ordered factorization index? That question asks what are the tuples called, whereas the present question asks what is the standard notation for writing the indexes.