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A215561
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Number A(n,k) of permutations of k indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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25
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 5, 30, 7, 1, 1, 1, 14, 420, 403, 35, 1, 1, 1, 42, 6930, 40350, 18720, 139, 1, 1, 1, 132, 126126, 5223915, 19369350, 746192, 1001, 1, 1, 1, 429, 2450448, 783353872, 27032968200, 9212531290, 71892912, 5701, 1
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OFFSET
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0,13
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COMMENTS
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"Late-growing permutations" were first defined by R. H. Hardin in A147681 and 18 related sequences. David Scambler observed that the set of orthogonal sequences includes A000108 and A007004, and he asked for the other orthogonal sequences, see link below.
"Early-growing permutations" with every partial sum >= the same partial sum averaged over all permutations define the same sequences.
Conjecture: Row r > 1 is asymptotic to c(r) * r^(r*n) / (Pi^((r-1)/2) * n^((r+1)/2)), where c(r) are a constants. - Vaclav Kotesovec, Sep 07 2016
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LINKS
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Alois P. Heinz, Antidiagonals n = 0..14, flattened
David Scambler et al., A147681 Late-growing permutations and follow-up messages on the SeqFan list, Aug 10 2012
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EXAMPLE
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A(2,2) = 2: (1,1,2,2), (1,2,1,2).
A(2,3) = 5: (1,1,1,2,2,2), (1,1,2,1,2,2), (1,1,2,2,1,2), (1,2,1,1,2,2), (1,2,1,2,1,2).
A(3,1) = 3: (1,2,3), (1,3,2), (2,1,3).
a(4,1) = 7: (1,2,3,4), (1,2,4,3), (1,3,2,4), (1,4,2,3), (2,1,3,4), (2,1,4,3), (2,3,1,4).
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 1, 2, 5, 14, 42, ...
1, 3, 30, 420, 6930, 126126, ...
1, 7, 403, 40350, 5223915, 783353872, ...
1, 35, 18720, 19369350, 27032968200, 44776592395920, ...
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MAPLE
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b:= proc(l) option remember; local m, n, g;
m, n:= nops(l), add(i, i=l);
g:= add(i*l[i], i=1..m)-(m+1)/2*(n-1);
`if`(n<2, 1, add(`if`(l[i]>0 and i<=g,
b(subsop(i=l[i]-1, l)), 0), i=1..m))
end:
A:= (n, k)-> b([k$n]):
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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b[l_] := b[l] = Module[{m, n, g}, {m, n} = {Length[l], Total[l]}; g = Sum[i*l[[i]], {i, 1, m}] - (m+1)/2*(n-1); If[n < 2, 1, Sum[If[l[[i]] > 0 && i <= g, b[ReplacePart[l, i -> l[[i]] - 1]], 0], {i, 1, m}]]]; a[n_, k_] := b[Array[k&, n]]; Table [Table [a[n, d-n], {n, 0, d}], {d, 0, 9}] // Flatten (* Jean-François Alcover, Dec 06 2013, translated from Maple *)
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CROSSREFS
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Columns k=0-19 give: A000012, A147681, A147682, A147684, A147686, A147687, A147692, A147694, A147695, A147697, A147698, A147700, A147705, A147707, A147712, A147713, A147714, A147715, A147717, A147769.
Rows n=0+1, 2-7 give: A000012, A000108, A007004, A215562, A215570, A215571, A215593.
Sequence in context: A106178 A305807 A205104 * A108714 A135508 A030413
Adjacent sequences: A215558 A215559 A215560 * A215562 A215563 A215564
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Aug 16 2012
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STATUS
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approved
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