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A051845
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Triangle T(n,k) read by rows, in which row n gives all permutations of digits 1..n interpreted in base n+1.
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8
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1, 5, 7, 27, 30, 39, 45, 54, 57, 194, 198, 214, 222, 238, 242, 294, 298, 334, 346, 358, 366, 414, 422, 434, 446, 482, 486, 538, 542, 558, 566, 582, 586, 1865, 1870, 1895, 1905, 1930, 1935, 2045, 2050, 2105, 2120, 2140, 2150, 2255, 2265, 2285, 2300, 2355, 2360
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OFFSET
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1,2
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COMMENTS
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All terms in any odd row 2m+1 are divisible by 2m+1
The n-th row has n! elements.
Variant of permutational numbers with shifted digits 0->1->2->...->p+1 in p+1 positional system -- see A134750. - Artur Jasinski, Nov 08 2007
All terms in any even-indexed row n=2m are divisible by m, where m>0.
Row n starts with T(n,1) = ((n+1)^(n+1)-n^2-n-1)/n^2 = A023811(n+1).
Row n ends with T(n,n!) = ((n+1)^(n+1)*(n-1)+1)/n^2 = A051846(n).
(End)
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LINKS
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EXAMPLE
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Triangle begins:
k=1 k=2 k=3 ...
n=1: 1;
n=2 5, 7;
n=3: 27, 30, ..., 57;
n=4: 194, 198, 214, ..., 586;
n=5: 1865, 1870, 1905, 1930, ..., 7465;
E.g., the permutations of digits 1, 2 and 3 in lexicographic order are 123, 132, 213, 231, 312, 321, which interpreted in base 4 give the third row of the table: 27, 30, 39, 45, 54, 57.
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MAPLE
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with(combinat, permute); compute_u_rows := proc(u) local a, n; a := []; for n from 1 to u do a := [op(a), op(map(list_in_base_b, permute(n), (n+1)))]; od; RETURN(a); end; list_in_base_b := proc(l, b) local k; add(l[nops(l)-k]*(b^k), k=0..(nops(l)-1)); end;
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MATHEMATICA
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a = {}; b = {}; Do[AppendTo[b, n]; w = Permutations[b]; Do[j = FromDigits[1 + w[[m]], n + 2]; AppendTo[a, j], {m, 1, Length[w]}], {n, 0, 5}]; a (* Artur Jasinski, Nov 08 2007 *)
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PROG
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(Python)
from itertools import permutations
def fd(d, b): return sum(di*b**i for i, di in enumerate(d[::-1]))
def row(n): return [fd(d, n+1) for d in permutations(range(1, n+1))]
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CROSSREFS
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KEYWORD
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easy,nonn,tabf,base
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AUTHOR
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STATUS
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approved
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