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A147679
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Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= n-1) is the number of permutations of [0..(n-1)] of spread k.
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1
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1, 1, 1, 0, 3, 3, 4, 8, 4, 8, 20, 25, 25, 25, 25, 144, 108, 108, 144, 108, 108, 630, 735, 735, 735, 735, 735, 735, 5696, 4608, 5248, 4608, 5696, 4608, 5248, 4608, 39366, 40824, 40824, 39285, 40824, 40824, 39285, 40824, 40824, 366400, 362000, 362000, 362000, 362000, 366400, 362000, 362000, 362000, 362000
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| The reference gives more terms, formulae, connection with A003112, etc.
s(pi):= Sum_{j=0..n-1} j*pi(j) (mod j) is defined as the spread of a permutation pi of [0..(n-1)].
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LINKS
| R. L. GRAHAM and D. H. LEHMER, ON THE PERMANENT OF SCHUR'S MATRIX, J. Australian Math. Soc., 21A (1976), 487-497.
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EXAMPLE
| Triangle begins:
1
1 1
0 3 3
4 8 4 8
20 25 25 25 25
144 108 108 144 108 108
...
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MAPLE
| b:= proc(n) option remember;
local l, p, r;
l:= array([i$i=0..n-1]);
r:= array([0$i=1..n]);
p:= proc(t, s)
local d, h, j;
if t=n then d:= ((s+(n-1)*l[n]) mod n) +1;
r[d]:= r[d]+1
else for j from t to n do
l[t], l[j]:= l[j], l[t];
p(t+1, (s+(t-1)*l[t]) )
od;
h:= l[t];
for j from t to n-1 do l[j]:= l[j+1] od;
l[n]:= h
fi
end;
p(1, 0);
eval(r)
end:
T:= (n, k)-> b(n)[k+1]:
seq (seq (T(n, k), k=0..n-1), n=1..10);
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PROG
| (Sage)
@CachedFunction
def A147679_row(n):
....row = [0]*n
....for p in Permutations(range(n)):
........spread = sum(i*px for i, px in enumerate(p)) % n
........row[spread] += 1
....return row
A147679 = lambda n, k: A147679_row(n)[k] # [D. S. McNeil, Dec 23 2010]
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CROSSREFS
| Cf. A003112. Row sums give: A000142. Columns k=0-3 give: A004204, A004205, A004206, A004246. Diagonal gives: A004205.
Sequence in context: A205116 A011959 A049927 * A137417 A137418 A155822
Adjacent sequences: A147676 A147677 A147678 * A147680 A147681 A147682
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KEYWORD
| nonn,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), May 01 2009
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EXTENSIONS
| Edited by Alois P. Heinz (heinz(AT)hs-heilbronn.de), Dec 22 2010
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