Mayan numerals

From the diagram given, the first group of $20$ numbers ( $0$ to $19$ ) has $40$ dots, $30$ bars, and $1$ shell for a total of $40 + 30 + 1 = 71$ symbols.

The next group of $20$ numbers ( $21$ to $39$ ) would have the same $71$ symbols but $1 \cdot 20$ extra dot symbols in the row above, the next group of $20$ numbers ( $40$ to $59$ ) would have the same $71$ symbols but $2 \cdot 20$ extra dot symbols in the row above, and so on, so that the first $20$ groups of $20$ numbers or first $400$ numbers ( $0$ to $399$ ) would have $( \text{total groups of } 20 \text{ in } 0 \text{ to } 399) \cdot 71 + (\text{total number of symbols for the numbers } 1 \text{ to } 19) \cdot 20$ $=$ $20 \cdot 71 + 70 \cdot 20 = 2820$ symbols.

The next group of $400$ numbers ( $400$ to $799$ ) would have the same $2820$ symbols as the first $400$ numbers but $20$ extra shell symbols for the zero place holders for the numbers from $400$ to $420$ and $1 \cdot 400$ extra dot symbols for the top row, for a total of $2820 + 20 + 1 \cdot 400 = 3240$ symbols. Similary, the numbers $800$ to $1199$ would have $2820 + 20 + 2 \cdot 400 = 3640$ symbols, the numbers $1200$ to $1599$ would have $2820 + 20 + 3 \cdot 400 = 4040$ symbols, and the numbers $1600$ to $1999$ would have $2820 + 20 + 4 \cdot 400 = 4440$ symbols.

Finally, the numbers $2000$ to $2019$ would have the same $71$ symbols as the numbers $0$ to $19$ , but $20$ extra shell symbols for the zero place holders and $20$ extra bar symbols on the top row for a total of $71 + 20 + 20 = 111$ symbols.

Therefore, the numbers $0$ to $2019$ would have a total of $2820 + 3240 + 3640 + 4040 + 4440 + 111 = 18291$ symbols. Since $0$ has $1$ symbol and $2019$ has $9$ symbols, the numbers $1$ to $2018$ would have a total of $18291 - 1 - 9 = \boxed{18281}$ symbols.