<abstract xmlns="http://www.w3.org/1999/xhtml">

Let <italic>π</italic><italic>n</italic> be a uniformly chosen random permutation on [<italic>n</italic>]. Using an analysis of the probability that two overlapping consecutive <italic>k</italic>-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns of all lengths <italic>k</italic> ∈ {1, 2,…, <italic>n</italic>} in <italic>π</italic><italic>n</italic> is <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_puma-2022-0031_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math>
<tex-math>{{{n^2}} \over 2}\left( {1 - o\left( 1 \right)} \right)</tex-math>
</alternatives>
</inline-formula> as <italic>n</italic> → ∞. This exhibits the fact that random permutations pack consecutive patterns near-perfectly.
</abstract>

The expected number of distinct consecutive patterns in a random permutation

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this note, we study the mean length of the longest increasing subsequence of a uniformly sampled involution that avoids the pattern 3412 and another pattern.
</abstract>

Houston, TX 77046, Departments of Mathematics and Engineering

The longest increasing subsequence in involutions avoiding 3412 and another pattern

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The expected number of distinct consecutive patterns in a random permutation

The longest increasing subsequence in involutions avoiding 3412 and another pattern

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