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

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

Author

R. H. Hardin, May 01 2009

Keywords

• nonn
• hard

Comments

Same as A145874.

Links

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

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

Extensions

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