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T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and the total absolute value of displacements not greater than 2*(n-1).
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%I #4 Oct 29 2015 07:45:14

%S 1,1,2,1,2,3,1,2,6,5,1,2,6,13,8,1,2,6,20,31,13,1,2,6,20,61,73,21,1,2,

%T 6,20,76,185,172,34,1,2,6,20,76,269,577,399,55,1,2,6,20,76,300,979,

%U 1788,932,89,1,2,6,20,76,300,1189,3552,5540,2177,144,1,2,6,20,76,300,1252

%N T(n,k)=Number of length n arrays of permutations of 0..n-1 with each element moved by -k to k places and the total absolute value of displacements not greater than 2*(n-1).

%C Table starts

%C ..1....1.....1.....1.....1.....1.....1......1......1......1......1......1

%C ..2....2.....2.....2.....2.....2.....2......2......2......2......2......2

%C ..3....6.....6.....6.....6.....6.....6......6......6......6......6......6

%C ..5...13....20....20....20....20....20.....20.....20.....20.....20.....20

%C ..8...31....61....76....76....76....76.....76.....76.....76.....76.....76

%C .13...73...185...269...300...300...300....300....300....300....300....300

%C .21..172...577...979..1189..1252..1252...1252...1252...1252...1252...1252

%C .34..399..1788..3552..4701..5197..5324...5324...5324...5324...5324...5324

%C .55..932..5540.13068.18754.21735.22869..23124..23124..23124..23124..23124

%C .89.2177.17165.47999.75243.91151.98497.101037.101548.101548.101548.101548

%H R. H. Hardin, <a href="/A263905/b263905.txt">Table of n, a(n) for n = 1..617</a>

%F Empirical for column k:

%F k=1: a(n) = a(n-1) +a(n-2)

%F k=2: a(n) = 2*a(n-1) +2*a(n-3) +a(n-4) -3*a(n-5) -2*a(n-7) +a(n-9)

%e Some solutions for n=7 k=4

%e ..3....0....3....2....4....0....3....0....0....1....4....0....3....0....2....1

%e ..0....3....1....1....1....1....1....5....1....3....0....2....0....2....1....3

%e ..2....1....5....5....0....3....0....3....4....2....2....1....4....4....4....2

%e ..6....4....2....0....3....2....6....4....5....6....1....5....2....1....0....6

%e ..1....5....0....3....6....4....4....1....6....0....5....6....5....3....6....0

%e ..4....6....4....4....2....5....2....2....2....4....6....4....1....6....5....5

%e ..5....2....6....6....5....6....5....6....3....5....3....3....6....5....3....4

%Y Column 1 is A000045(n+1).

%K nonn,tabl

%O 1,3

%A _R. H. Hardin_, Oct 29 2015