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A152883 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which k is an excedance (n >= 2, 1 <= k <= n-1). An excedance of a permutation p is a value j such that p(j) > j.

%I A152883 #12 Jun 13 2021 08:46:14
%S A152883 1,4,2,18,12,6,96,72,48,24,600,480,360,240,120,4320,3600,2880,2160,
%T A152883 1440,720,35280,30240,25200,20160,15120,10080,5040,322560,282240,
%U A152883 241920,201600,161280,120960,80640,40320,3265920,2903040,2540160,2177280,1814400,1451520,1088640,725760,362880
%N A152883 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} in which k is an excedance (n >= 2, 1 <= k <= n-1). An excedance of a permutation p is a value j such that p(j) > j.
%C A152883 Sum of entries in row n = n!*(n-1)/2 = A001286(n) (the Lah numbers).
%C A152883 T(n,n-1) = (n-1)! (A000142).
%F A152883 T(n,k) = (n-1)!*(n-k) (n >= 2, 1 <= k <= n-1). [Proof: n-k choices for p(k) and (n-1)! choices for the remaining entries of p.]
%e A152883 T(4,3) = 6 because the permutations of {1,2,3,4} in which 3 is an excedance are 1243, 1342, 3142, 2143, 2341 and 3241.
%e A152883 Triangle starts:
%e A152883     1;
%e A152883     4,   2;
%e A152883    18,  12,   6;
%e A152883    96,  72,  48,  24;
%e A152883   600, 480, 360, 240, 120;
%p A152883 T := proc (n, k) options operator, arrow: factorial(n-1)*(n-k) end proc: for n from 2 to 10 do seq(T(n, k), k = 1 .. n-1) end do;
%Y A152883 Cf. A000142, A001286.
%K A152883 nonn,tabl
%O A152883 2,2
%A A152883 _Emeric Deutsch_, Jan 13 2009