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
Examples
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
Links
- Alois P. Heinz, Rows n = 1..141, flattened
- The Riddler Express, O. Roeder, solution to 2018-Aug 17 problem by K. Hudson
Crossrefs
T(n+1,1) = A144188(n).
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.
Comments