A244148 The number of ways one can assign values to n arrays a_{1},...,a_{n} of increasing size (size of a_{1} is 1, size of a_{2} is 2, ..., size of a_{n} is n) using the numbers 1, ..., n*(n+1)/2, distinctly, such that the positions of array a_{i} can only be assigned values in the interval ((n+1)-i),... , (n*(n+1)/2-(n-i)).

1, 2, 72, 115200, 13276569600, 165253252792320000, 312379127174190543667200000, 120053472861445542607502662277529600000, 12098873398276702490569569159619238449643520000000000, 400639807706466477973460949403651522366500906696560470917120000000000

Offset: 1

Author

David M. Cerna, Jun 21 2014

Keywords

• easy
• nonn

Comments

This sequence provides an upper bound for the following sequence: the number of ways one can assign values to n arrays a_{1},...,a_{n} of increasing size (size of a_{1} is 1, size of a_{2} is 2, ..., size of a_{n} is n) using the numbers 1, ..., n*(n+1)/2, distinctly, such that for the j^th position of array a_{i} (a_{i}(j)) one of the follow holds, a_{i+1}(j+1) < a_{i}(j) < a_{i+1}(j) or a_{i+1}(j) < a_{i}(j) < a_{i+1}(j+1). Currently, there is no formula known for enumerating this sequence.

Links

• David M. Cerna, Table of n, a(n) for n = 1..50
• David M. Cerna, Proof of enumeration formula
• Clark Kimberling, Unsolved Problems and Rewards: Number 18

Programs

PARI

a(n)=prod(k=1,n,k!* binomial((n^2 - 3*n + 5*k - k^2)/2 , k)); \\ Joerg Arndt, Jun 22 2014

Formula

a(n) = Prod_{k=1..n} (k!* binomial((n^2 - 3*n + 5*k - k^2)/2 , k)).