#Pyth, <s>64</s> <s>48</s> 50 bytes

    #jdhfqzsTsm^+msXtjCk2U2"<>""
    \r.1"",<..>,"dlzB

[Test case.][1]

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Version that doesn't take forever ($O(9^{n/3}$) [here][2], in 52 bytes.

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This is the brute force approach, generate all sequences and check whether any sum to the input. Fish diagrams compressed as characters, whose binary representations are the `>` and `<`. The whole thing is wrapped in a try-catch block so that no output occurs when no results are found.

This is an $O(9^n)$ solution.

Some characters are stripped above, because control characters are used. They are reproduced faithfully at the link above.



xxd output:

    0000000: 236a 6468 6671 7a73 5473 6d5e 2b6d 7358  #jdhfqzsTsm^+msX
    0000010: 746a 436b 3255 3222 3c3e 2222 0a5c 7212  tjCk2U2"<>"".\r.
    0000020: 141b 1d2e 3122 222c 3c2e 2e3e 2c22 646c  ....1"",<..>,"dl
    0000030: 7a42                                     zB


  [1]: https://pyth.herokuapp.com/?code=jdhfqzsTsm%5E%2BmsXtjCk2U2%22%3C%3E%22%22%0A%5Cr%12%14%1B%1D.1%22%22%2C%3C..%3E%2C%22dlz&input=%3C%3E%3C%3E%3C%3E
  [2]: https://pyth.herokuapp.com/?code=%23jdhfqzsTsm%5E%2BmsXtjCk2U2%22%3C%3E%22%22%0A%5Cr%12%14%1B%1D.1%22%22%2C%3C..%3E%2C%22dlcz3B&input=%3C%3E%3C%3C%3C%3E%3C&debug=0