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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


%S 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,

%T 178,224,286,1519,1233,1009,831,831,1009,1233,1519,9184,7665,6432,

%U 5423,4592,5423,6432,7665,9184,62000,52816,45151,38719,33296,33296,38719,45151,52816,62000

%N 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).

%C 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.

%H Alois P. Heinz, <a href="/A318354/b318354.txt">Rows n = 1..141, flattened</a>

%H The Riddler Express, O. Roeder, <a href="https://fivethirtyeight.com/features/how-many-hoops-will-kids-jump-through-to-play-rock-paper-scissors/">solution to 2018-Aug 17 problem by K. Hudson</a>

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

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

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

%e 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.

%e The triangle begins:

%e 1

%e 1 1

%e 2 1 2

%e 5 3 3 5

%e 16 11 8 11 16

%e 62 46 35 35 46 62

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

%K nonn,tabl

%O 1,4

%A _Glen Whitney_, Aug 24 2018

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Last modified December 2 05:01 EST 2021. Contains 349437 sequences. (Running on oeis4.)