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A181372 Square array read by antidiagonals: a(p,n) is the number of inversions in all p-ary words of length n on {0,1,2,...,p-1} (p>=2, n>=2).

%I A181372 #2 Mar 30 2012 17:36:24
%S A181372 1,3,6,6,27,24,10,72,162,80,15,150,576,810,240,21,270,1500,3840,3645,
%T A181372 672,28,441,3240,12500,23040,15309,1792,36,672,6174,32400,93750,
%U A181372 129024,61236,4608,45,972,10752,72030,291600,656250,688128,236196,11520
%N A181372 Square array read by antidiagonals: a(p,n) is the number of inversions in all p-ary words of length n on {0,1,2,...,p-1} (p>=2, n>=2).
%C A181372 a(2,n)=A001788(n-1).
%C A181372 a(3,n)=A129530(n).
%C A181372 a(4,n)=A129532(n).
%F A181372 a(p,n)=n(n-1)(p-1)p^{n-1}/4.
%e A181372 a(2,3)=6 because the words 010, 100, 101, 110 have 1, 2, 1, 2 inversions, respectively, while 000, 001, 011 and 111 have none.
%e A181372 a(3,2)=3 because each of the words 10, 20, 21 has 1 inversion, while 00, 01, 02, 11, 12, 22 have none.
%e A181372 Square array starts:
%e A181372 1,6,24,80,240, ...;
%e A181372 3,27,162,810,3645, ... ;
%e A181372 6,72,576, 3840,23040, ... ;
%e A181372 10,150,1500,12500, 93750,...;
%p A181372 a := proc (p, n) options operator, arrow: (1/4)*n*(n-1)*(p-1)*p^(n-1) end proc: for p from 2 to 8 do seq(a(p, n), n = 2 .. 8) end do; # yields the first seven rows and seven columns of the square array
%Y A181372 Cf. A001788, A129530, A129532.
%K A181372 nonn,tabl
%O A181372 2,2
%A A181372 _Emeric Deutsch_, Oct 31 2010