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Omni-MATH / For any finite sequence of positive integers …S(π) be the number of strictly increasing subsequences in …π={3,1,2,4}, there are five increasing sub-sequences: …

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problem

For any finite sequence of positive integers \pi, let S(π)S(\pi) be the number of strictly increasing subsequences in \pi with length 2 or more. For example, in the sequence π={3,1,2,4}\pi=\{3,1,2,4\}, there are five increasing sub-sequences: {3,4},{1,2},{1,4},{2,4}\{3,4\},\{1,2\},\{1,4\},\{2,4\}, and {1,2,4}\{1,2,4\}, so S(π)=5S(\pi)=5. In an eight-player game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order \pi from left to right in her hand. Determine ∑πS(π)\sum_{\pi} S(\pi) where the sum is taken over all possible orders \pi of the card values.
Plain-text mathematical notation (without MathML)
For any finite sequence of positive integers \pi, let S(π) be the number of strictly increasing subsequences in \pi with length 2 or more. For example, in the sequence π={3,1,2,4}, there are five increasing sub-sequences: {3,4},{1,2},{1,4},{2,4}, and {1,2,4}, so S(π)=5. In an eight-player game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order \pi from left to right in her hand. Determine ∑_(π)S(π) where the sum is taken over all possible orders \pi of the card values.
Original LaTeX notation
For any finite sequence of positive integers \pi, let $S(\pi)$ be the number of strictly increasing subsequences in \pi with length 2 or more. For example, in the sequence $\pi=\{3,1,2,4\}$, there are five increasing sub-sequences: $\{3,4\},\{1,2\},\{1,4\},\{2,4\}$, and $\{1,2,4\}$, so $S(\pi)=5$. In an eight-player game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order \pi from left to right in her hand. Determine $\sum_{\pi} S(\pi)$ where the sum is taken over all possible orders \pi of the card values.

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Source and history

task_id
30bf57d2-0c5f-5ea8-b7e3-f4602bbd4cc9
task_key
test--30bf57d2-0c5f-5ea8-b7e3-f4602bbd4cc9
task_revision_id
1
config / split
/ test

Official source

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