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%I #34 Jan 02 2023 12:30:48
%S 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,
%T 42,6930,40350,18720,139,1,1,1,132,126126,5223915,19369350,746192,
%U 1001,1,1,1,429,2450448,783353872,27032968200,9212531290,71892912,5701,1
%N 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.
%C "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.
%C "Early-growing permutations" with every partial sum >= the same partial sum averaged over all permutations define the same sequences.
%C 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
%H Alois P. Heinz, <a href="/A215561/b215561.txt">Antidiagonals n = 0..14, flattened</a>
%H David Scambler et al., <a href="http://list.seqfan.eu/oldermail/seqfan/2012-August/010018.html">A147681 Late-growing permutations</a> and follow-up messages on the SeqFan list, Aug 10 2012
%e A(2,2) = 2: (1,1,2,2), (1,2,1,2).
%e 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).
%e A(3,1) = 3: (1,2,3), (1,3,2), (2,1,3).
%e 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).
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 2, 5, 14, 42, ...
%e 1, 3, 30, 420, 6930, 126126, ...
%e 1, 7, 403, 40350, 5223915, 783353872, ...
%e 1, 35, 18720, 19369350, 27032968200, 44776592395920, ...
%p b:= proc(l) option remember; local m, n, g;
%p m, n:= nops(l), add(i, i=l);
%p g:= add(i*l[i], i=1..m)-(m+1)/2*(n-1);
%p `if`(n<2, 1, add(`if`(l[i]>0 and i<=g,
%p b(subsop(i=l[i]-1, l)), 0), i=1..m))
%p end:
%p A:= (n, k)-> b([k$n]):
%p seq(seq(A(n, d-n), n=0..d), d=0..10);
%t 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 *)
%Y 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.
%Y Rows n=0+1, 2-7 give: A000012, A000108, A007004, A215562, A215570, A215571, A215593.
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Aug 16 2012
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