A318354 Triangle read by rows: T(n,k) is the number of permutations p of {1..n} such that p(1)=k and p(i+1) < p(i) iff a strict majority of {1..n} \ {p(1)..p(i)} are < p(i).

1, 1, 1, 2, 1, 2, 5, 3, 3, 5, 16, 11, 8, 11, 16, 62, 46, 35, 35, 46, 62, 286, 224, 178, 143, 178, 224, 286, 1519, 1233, 1009, 831, 831, 1009, 1233, 1519, 9184, 7665, 6432, 5423, 4592, 5423, 6432, 7665, 9184, 62000, 52816, 45151, 38719, 33296, 33296, 38719, 45151, 52816, 62000

Offset

1,4

Comments

If you shuffle n cards numbered 1 to n and then turn them over one at a time, guessing whether the next will be larger than the previous by the (optimal) rule that you guess "larger" unless there are more cards remaining smaller than the one just revealed, T(n,k) is the number of arrangements such that the first card revealed is k and you guess correctly every time.

Links

Alois P. Heinz, Rows n = 1..141, flattened

The Riddler Express, O. Roeder, solution to 2018-Aug 17 problem by K. Hudson

Formula

For k <= n/2 + 1: T(n+1,k) = Sum_{i=k..n} T(n,i);

For k >= n/2 + 1: T(n+1,k) = Sum_{i=1..k-1} T(n,i).

T(n+1,k+1) = f(n,k), where f(n,k) is the auxiliary function defined in the formula for A144188.

Example

Suppose you are playing with four cards and you initially turn over a "2". You guess "larger" because there are two larger cards, 3 and 4, remaining, and only 1 smaller card, 1, remaining. You continue playing in this way, guessing larger unless there are (strictly) more smaller cards remaining. You guess correctly every time if the order of the cards was 2,3,4,1; 2,4,3,1; or 2,4,1,3. Thus T(4,2) = 3.
The triangle begins:
        1
       1 1
      2 1 2
     5 3 3 5
  16 11 8 11 16
62 46 35 35 46 62

Crossrefs

T(n+1,1) = A144188(n).

Sequence in context: A345278 A212431 A346517 * A380079 A348373 A106480

Adjacent sequences: A318351 A318352 A318353 * A318355 A318356 A318357

Keyword

nonn,tabl

Author

Glen Whitney, Aug 24 2018

Status

approved