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A350227 Triangular array read by rows. T(n,k) is the number of partial permutations on [n] with exactly k connected components, n>=0, 0<=k<=n.

%I A350227 #9 Dec 20 2021 18:16:26
%S A350227 1,0,2,0,3,4,0,8,18,8,0,30,91,72,16,0,144,540,590,240,32,0,840,3718,
%T A350227 5085,2900,720,64,0,5760,29232,47516,34230,12040,2016,128,0,45360,
%U A350227 258732,484092,416857,186480,44576,5376,256,0,403200,2547360,5368184,5340888,2869314,876960,151872,13824,512
%N A350227 Triangular array read by rows.  T(n,k) is the number of partial permutations on [n] with exactly k connected components, n>=0, 0<=k<=n.
%F A350227 T(n,n) = 2^n = A000079(n) (counts the idempotent elements).
%F A350227 For n>=1, T(n,1) = (n-1)! + n! = A001048(n) (the component is a cycle or a directed path to a point with a self loop).
%F A350227 E.g.f.: exp(y*log(1/(1-x)))*exp(y*x/(1-x)).
%e A350227 Triangle begins:
%e A350227   1;
%e A350227   0,   2;
%e A350227   0,   3,   4;
%e A350227   0,   8,  18,   8;
%e A350227   0,  30,  91,  72,  16;
%e A350227   0, 144, 540, 590, 240, 32;
%e A350227   ...
%t A350227 nn = 9; Table[Take[(Range[0, nn]! CoefficientList[Series[1/(1 - x)^y Exp[y x/(1 - x)], {x, 0, nn}], {x, y}])[[i, All]], i], {i, 1, nn + 1}] // Grid
%Y A350227 Cf. A000079, A132393, A001048, A002720 (row sums).
%K A350227 nonn,tabl
%O A350227 0,3
%A A350227 _Geoffrey Critzer_, Dec 20 2021