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A374738

Table read by ascending antidiagonals: T(m,n) = number of (n-1)-metered (m,n)-parking functions.

1, 1, 2, 1, 3, 3, 1, 4, 8, 4, 1, 6, 16, 15, 5, 1, 8, 27, 50, 24, 6, 1, 12, 48, 125, 108, 35, 7, 1, 16, 96, 257, 432, 196, 48, 8, 1, 24, 162, 540, 1296, 1029, 320, 63, 9, 1, 32, 288, 1200, 3156, 4802, 2048, 486, 80, 10, 1, 48, 576, 3000, 7734, 16807, 12288, 3645, 700, 99, 11

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OFFSET

1,3

LINKS

Table of n, a(n) for n=1..66.

Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024.

FORMULA

T(n+k,n) = Sum_{sigma = (sigma_1, ..., sigma_n) in S_n} (( Product_{i=1..n} L_{i}(sigma))( Product_{j=1..k} sigma_j mod n )), where k>0 and L_{i}(sigma) is the largest index h with i<h for which sigma_i >= sigma_N for all N in {i-j, i-j+1, ..., i-1, i}.

EXAMPLE

Table begins:

1, 2, 3, 4, 5, 6, 7, ...

1, 3, 8, 15, 24, 35, 48, ...

1, 4, 16, 50, 108, 196, 320, ...

1, 6, 27, 125, 432, 1029, 2048, ...

1, 8, 48, 257, 1296, 4802, 12288, ...

1, 12, 96, 540, 3156, 16807, 65536, ...

1, 16, 162, 1200, 7734, 47442, 262144, ...

...

CROSSREFS

The n=m+1 diagonal is A007334.