A372821 Table read by antidiagonals: T(m,n) = number of (m-2)-metered (m,n)-parking functions.
0, 1, 0, 0, 4, 0, 0, 4, 9, 0, 0, 0, 21, 16, 0, 0, 0, 27, 56, 25, 0, 0, 0, 0, 163, 115, 36, 0, 0, 0, 0, 257, 483, 204, 49, 0, 0, 0, 0, 0, 1686, 1095, 329, 64, 0, 0, 0, 0, 0, 3156, 5367, 2131, 496, 81, 0, 0, 0, 0, 0, 0, 21858, 13076, 3747, 711, 100, 0, 0, 0, 0, 0, 0, 47442, 73276, 27309, 6123, 980, 121, 0
Offset: 1
Examples
Table begins: 0, 0, 0, 0, 0, 0, 0, ... 1, 4, 9, 16, 25, 36, 49, ... 0, 4, 21, 56, 115, 204, 329, ... 0, 0, 27, 163, 483, 1095, 2131, ... 0, 0, 0, 257, 1686, 5367, 13076, ... 0, 0, 0, 0, 3156, 21858, 73276, ... 0, 0, 0, 0, 0, 47442, 341192, ... ...
Links
- Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024.
Formula
T(m,n) = (n-m+2)^2*(m-1)^(m-3) + Sum_{k=n-m+3...n} binomial(m-2, n-k)*(n-k+1)^(n-k-1)*[binomial(k+1,2)*(n+m+2)*k^(m-n+k-3) + (k*(n-m+1) - binomial(n-m+2,2))*(k-n+m-1)^(k-n+m-3) + Sum_{j=n-m+2} (jk - binomial(j+1,2))*binomial(m-2-n+k, k-1-j)*(n-m+1)*j^(j+m-2-n)*(k-j)^(k-j-2)].