A three member sequence $$a, b, c$$ is said to be an up-down sequence if $$a c$$. For example $$1, 3, 2$$ is an up-down sequence. The sequence 1342 contains three up-down sequences, namely $$(1,3,2)$$, $$(1,4,2)$$ and $$(3,4,2)$$. How many up-down sequences are contained in the sequence 132597684?
Solution
Take each digit as the peak and multiply the number of smaller digits before it by the number of smaller digits after it. For the digits $$1, 3, 2, 5, 9, 7, 6, 8, 4$$ these products are $$0, 1 \times 1, 1 \times 0, 3 \times 1, 4 \times 4, 4 \times 2, 4 \times 1, 6 \times 1, 0$$. Adding them gives $$1+0+3+16+8+4+6 = 38$$.
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