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%I #13 Jan 18 2022 06:01:18
%S 1,1,1,1,4,2,2,1,9,9,18,3,18,6,1,16,24,72,16,144,96,4,48,36,144,24,1,
%T 25,50,200,50,600,600,25,400,300,1800,600,5,100,200,600,900,1200,120,
%U 1,36,90,450,120,1800,2400,90,1800,1350,10800,5400,36,900,1800,7200,10800,21600,4320,6,180,450,1800,300,7200,7200,1800,16200,10800,720
%N Irregular triangle read by rows: T(n,k) is the number of partial functions on [n] such that the sizes of the preimages of the individual elements in the range form the k-th partition in the class of all partitions listed in Abramowitz and Stegun order, n>=0, 0<=k<=A000070(n).
%C The last A000041(n) entries of each row give A049009.
%C Row sums are (n+1)^n = A000169(n+1).
%e 1,
%e 1, 1,
%e 1, 4, 2, 2,
%e 1, 9, 9, 18, 3, 18, 6,
%e 1, 16, 24, 72, 16, 144, 96, 4, 48, 36, 144, 24
%t g[n_, list_] := Multinomial @@ Join[{n - Length[list]}, Table[Count[list, i], {i, 1, n}]]* Multinomial @@ Join[{n - Total[list]}, list]; Table[Map[g[nn, #] &,
%t Level[Table[IntegerPartitions[k], {k, 0, nn}], {2}]], {nn, 0, 5}] // Grid
%Y Cf. A000169, A049009, A000041, A000070.
%K nonn,tabf
%O 0,5
%A _Geoffrey Critzer_, Jan 16 2022