Map a list of indefinite size to a number!
Jelly, 18 16 bytes

List to integer, 10 8 bytes

TṪạL;³ÆẸ


Maps lists of non-negative integers to positive integers. Try it online!

Integer to list, 8 bytes

ÆE©Ḣ0ẋ®;


Maps positive integers to lists of non-negative integers . Try it online!

Background

Let p0, p1, p2, ⋯ be the sequence of prime numbers in ascending order. 

For each list of non-negative integers A := [a1, ⋯, an], we map A to p0z(A)p1a1⋯pnan, where z(A) is the number of trailing zeroes of A.

Reversing the above map in straightforward. For a positive integer k, we factorize it uniquely as the product of consecutive prime powers n = p0α0p1α1⋯pnαn, where αn > 0, then reconstruct the list as [α1, ⋯, αn], appending α0 zeroes.

How it works

List to integer

TṪạL;³ÆẸ  Main link. Argument: A (list of non-negative integers)

T         Yield all indices of A that correspond to truthy (i.e., non-zero) items.
 Ṫ        Tail; select the last truthy index.
          This returns 0 if the list is empty.
   L      Yield the length of A.
  ạ       Compute the absolute difference of the last truthy index and the length.
          This yields the amount of trailing zeroes of A.
    ;³    Prepend the difference to A.
      ÆẸ  Convert the list from prime exponents to integer.


Integer to list

ÆE©Ḣ0ẋ®;  Main link. Input: k (positive integer)

ÆE        Convert k to the list of its prime exponents.
  ©       Save the list of prime exponents in the register.
   Ḣ      Head; pop the first exponent.
          If the list is empty, this yields 0.
    0ẋ    Construct a list of that many zeroes.
      ®;  Concatenate the popped list of exponents with the list of zeroes.       


Example output

The first one hundred positive integers map to the following lists.

  1: []
  2: [0]
  3: [1]
  4: [0, 0]
  5: [0, 1]
  6: [1, 0]
  7: [0, 0, 1]
  8: [0, 0, 0]
  9: [2]
 10: [0, 1, 0]
 11: [0, 0, 0, 1]
 12: [1, 0, 0]
 13: [0, 0, 0, 0, 1]
 14: [0, 0, 1, 0]
 15: [1, 1]
 16: [0, 0, 0, 0]
 17: [0, 0, 0, 0, 0, 1]
 18: [2, 0]
 19: [0, 0, 0, 0, 0, 0, 1]
 20: [0, 1, 0, 0]
 21: [1, 0, 1]
 22: [0, 0, 0, 1, 0]
 23: [0, 0, 0, 0, 0, 0, 0, 1]
 24: [1, 0, 0, 0]
 25: [0, 2]
 26: [0, 0, 0, 0, 1, 0]
 27: [3]
 28: [0, 0, 1, 0, 0]
 29: [0, 0, 0, 0, 0, 0, 0, 0, 1]
 30: [1, 1, 0]
 31: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 32: [0, 0, 0, 0, 0]
 33: [1, 0, 0, 1]
 34: [0, 0, 0, 0, 0, 1, 0]
 35: [0, 1, 1]
 36: [2, 0, 0]
 37: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 38: [0, 0, 0, 0, 0, 0, 1, 0]
 39: [1, 0, 0, 0, 1]
 40: [0, 1, 0, 0, 0]
 41: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 42: [1, 0, 1, 0]
 43: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 44: [0, 0, 0, 1, 0, 0]
 45: [2, 1]
 46: [0, 0, 0, 0, 0, 0, 0, 1, 0]
 47: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 48: [1, 0, 0, 0, 0]
 49: [0, 0, 2]
 50: [0, 2, 0]
 51: [1, 0, 0, 0, 0, 1]
 52: [0, 0, 0, 0, 1, 0, 0]
 53: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 54: [3, 0]
 55: [0, 1, 0, 1]
 56: [0, 0, 1, 0, 0, 0]
 57: [1, 0, 0, 0, 0, 0, 1]
 58: [0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
 59: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 60: [1, 1, 0, 0]
 61: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 62: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
 63: [2, 0, 1]
 64: [0, 0, 0, 0, 0, 0]
 65: [0, 1, 0, 0, 1]
 66: [1, 0, 0, 1, 0]
 67: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 68: [0, 0, 0, 0, 0, 1, 0, 0]
 69: [1, 0, 0, 0, 0, 0, 0, 1]
 70: [0, 1, 1, 0]
 71: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 72: [2, 0, 0, 0]
 73: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 74: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
 75: [1, 2]
 76: [0, 0, 0, 0, 0, 0, 1, 0, 0]
 77: [0, 0, 1, 1]
 78: [1, 0, 0, 0, 1, 0]
 79: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 80: [0, 1, 0, 0, 0, 0]
 81: [4]
 82: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
 83: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 84: [1, 0, 1, 0, 0]
 85: [0, 1, 0, 0, 0, 1]
 86: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
 87: [1, 0, 0, 0, 0, 0, 0, 0, 1]
 88: [0, 0, 0, 1, 0, 0, 0]
 89: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 90: [2, 1, 0]
 91: [0, 0, 1, 0, 1]
 92: [0, 0, 0, 0, 0, 0, 0, 1, 0, 0]
 93: [1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 94: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
 95: [0, 1, 0, 0, 0, 0, 1]
 96: [1, 0, 0, 0, 0, 0]
 97: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
 98: [0, 0, 2, 0]
 99: [2, 0, 0, 1]
100: [0, 2, 0, 0]