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%I #11 Mar 07 2021 17:30:11
%S 1,1,1,1,3,1,0,0,1,1,6,4,4,0,4,0,3,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,
%T 10,10,20,1,20,0,15,0,6,0,15,0,0,4,0,0,0,0,4,0,0,0,5,0,0,0,0,0,4,0,0,
%U 0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n!) = number of permutations on n letters whose fiber under the parking function map phi has size k.
%H Laura Colmenarejo, Pamela E. Harris, Zakiya Jones, Christo Keller, Andrees Ramos Rodriguez Eunice Sukarto and Andres R. Vindas-Melendez, <a href="http://ecajournal.haifa.ac.il/Volume2021/ECA2021_S2A11.pdf">Counting k-Naples Parking Functions Through Permutations and the k-Naples Area Statistic</a>, Enumerative Combinatorics and Applications, ECA 1:2 (2021) Article S2R11. See page 9. Proposition 4.1 gives a recurrence.
%e Triangle begins:
%e 1,
%e 1,1,
%e 1,3,1,0,0,1,
%e 1,6,4,4,0,4,0,3,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,
%e ...
%e Row 3 is 1,3,1,0,0,1 as F3(q) = q + 3q^2 + q^3 + q^6 = q + 3q^2 + q^3 + 0q^4 + 0 q^5 + q^6. k in name are exponents in the power q^k and terms in rows are coefficients. The row lists the coefficients starting at k = 1 and ending at k = n!. - _David A. Corneth_, Mar 07 2021
%K nonn,tabf
%O 1,5
%A _N. J. A. Sloane_, Mar 07 2021