A333829 Triangle read by rows: T(n,k) is the number of parking functions of length n with k strict descents. T(n,k) for n >= 1 and 0 <= k <= n-1.
1, 2, 1, 5, 10, 1, 14, 73, 37, 1, 42, 476, 651, 126, 1, 132, 2952, 8530, 4770, 422, 1, 429, 17886, 95943, 114612, 31851, 1422, 1, 1430, 107305, 987261, 2162033, 1317133, 202953, 4853, 1, 4862, 642070, 9613054, 35196634, 39471355, 13792438, 1262800, 16786, 1
Offset: 1
Examples
The triangle T(n,k) begins: n/k 0 1 2 3 4 5 1 1 2 2 1 3 5 10 1 4 14 73 37 1 5 42 476 651 126 1 6 132 2952 8530 4770 422 1 ... The 10 parking functions of length 3 with 1 strict descent are: [[1, 2, 1], [2, 1, 1], [1, 3, 1], [3, 1, 1], [2, 1, 2], [2, 2, 1], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2]].
Links
- Ari Cruz, Pamela E. Harris, Kimberly J. Harry, Jan Kretschmann, Matt McClinton, Alex Moon, John O. Museus, and Eric Redmon, On some discrete statistics of parking functions, arXiv:2312.16786 [math.CO], 2023.
- Paul R. F. Schumacher, Descents in Parking Functions, J. Int. Seq. 21 (2018), #18.2.3.
Programs
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SageMath
var('z,t') assume(0 -
SageMath
# using[latte_int from LattE] # Install with "sage -i latte_int". # Another method is to compute the Ehrhart h^*-polynomial of a polytope. var('z,t') def Tpol(n): p = Polyhedron( NonDecreasingParkingFunctions(n+1) ).ehrhart_polynomial() return expand(factor( (1-z)**(n+1) * sum( p * z**t , t , 0 , oo ) )) def T(n,k): return Tpol(n).list()[n-1-k]
Comments