

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, 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
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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 2018Aug 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..k1} 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 * A348373 A106480 A099602
Adjacent sequences: A318351 A318352 A318353 * A318355 A318356 A318357


KEYWORD

nonn,tabl


AUTHOR

Glen Whitney, Aug 24 2018


STATUS

approved



