A147692 -id:A147692 - cp's OEIS frontend

A215561 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.

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

Offset: 0

Alois P. Heinz, Aug 16 2012

"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

			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, ...

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

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.

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);

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 *)

A147681 Late-growing permutations: number of permutations of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 1, 1, 3, 7, 35, 139, 1001, 5701, 53109, 402985, 4605271, 43665667, 589809987, 6735960079, 104899483845, 1402547616085, 24698838710457, 378845419610773, 7444522779300351, 128830635114146047, 2792467448952670671, 53854927962971227495, 1276369340371154144337, 27141331409803338993193, 698008560075731437652425, 16228797258964121571885457, 450111715263775132783135875

Offset: 0

R. H. Hardin, May 01 2009

Same as A145874.

David Scambler et al., A147681 Late-growing permutations and follow-up messages on the SeqFan list, Aug 10 2012

This is the first of 19 related sequences, the others being A147682, A147684, A147686, A147687, A147692, A147694, A147695, A147697, A147698, A147700, A147705, A147707, A147712, A147713, A147714, A147715, A147717, A147769.

Column k=1 of A215561.

a:= proc(n) option remember; local b, m; m:= n*(n+1)/2;
      b:= proc(s) option remember; local h, g; h:= nops(s);
            g:= (n-h+1)*(1+n)/2 -m +add(i, i=s); `if`(h<2, 1,
            add(`if`(s[i]<=g, b(subsop(i=NULL, s)), 0), i=1..h))
          end; forget(b);
      b([$1..n])
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 10 2012

a[n_] := a[n] = Module[{b, m}, m = n*(n+1)/2; b[s_List] := b[s] = Module[{h, g}, h = Length[s]; g = (n-h+1)*(1+n)/2 - m + Total[s]; If[h<2, 1, Sum[If[s[[i]] <= g, b[ReplacePart[s, i -> Sequence[]]], 0], {i, 1, h}]]];  b[Range[n]]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

a(22) from Alois P. Heinz, Aug 10 2012
a(23) from Alois P. Heinz, Nov 01 2014
a(24)-a(25) from Vaclav Kotesovec, Jan 31 2015
a(26)-a(27) from Vaclav Kotesovec, Sep 07 2016

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A215561 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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