A147681 -id:A147681 - 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 *)

A147682 Late-growing permutations: number of permutations of 2 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 1, 2, 30, 403, 18720, 746192, 71892912, 5873837638, 951265850580, 133244998049858, 32484245570649180, 6956417433946216990, 2375465385671586163800, 723157816455776560763294, 329255781245519867317200240, 135189844328107458501296074066, 79079768375837127458516103725820

Offset: 0

R. H. Hardin, May 01 2009

Cf. A147681.

Column k=2 of A215561.

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-> b([2$n]):
seq(a(n), n=1..10);  # Alois P. Heinz, Aug 16 2012

b[l_List] := 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_] := b[Table[2, {n}]]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

a(14) from Alois P. Heinz, Aug 16 2012
a(15) from Alois P. Heinz, Nov 02 2014
a(16) from Vaclav Kotesovec, Sep 07 2016
a(17) from Vaclav Kotesovec, Sep 08 2016

A147684 Late-growing permutations: number of permutations of 3 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 1, 5, 420, 40350, 19369350, 9212531290, 13126885205000, 17810026933803520, 55560479543584645500, 164686892656273830526336, 953018107457232657556038400

Offset: 0

R. H. Hardin, May 01 2009

nonn

Cf. A147681.

Column k=3 of A215561.

a(0), a(11) from Alois P. Heinz, Nov 01 2014

A147686 Late-growing permutations: number of permutations of 4 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 14, 6930, 5223915, 27032968200, 164401445439455, 3627155158988429250, 86733224358763671877835, 5469038805616093755410863500

Offset: 1

R. H. Hardin May 01 2009

nonn

Cf. A147681.

Column k=4 of A215561.

A147687 Late-growing permutations: number of permutations of 5 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 42, 126126, 783353872, 44776592395920, 3611684199828856072, 1267664556730792079292048, 551951935901513814954541886968

Offset: 1

R. H. Hardin May 01 2009

nonn

Cf. A147681.

Column k=5 of A215561.

A147692 Late-growing permutations: number of permutations of 6 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 132, 2450448, 129141898872, 82881380383401600, 90695437030756958966384, 515544601327354412382720479328, 4172457328749067883416103335334343279

Offset: 1

R. H. Hardin May 01 2009

nonn

Cf. A147681.

Column k=6 of A215561.

a(8) from Alois P. Heinz, Aug 17 2012

A147694 Late-growing permutations: number of permutations of 7 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 429, 49884120, 22745605840236, 165850226337286576800, 2500267880518574604245088816, 233099041543988273824859604028713600

Offset: 1

R. H. Hardin May 01 2009

nonn

Cf. A147681.

Column k=7 of A215561.

A147695 Late-growing permutations: number of permutations of 8 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 1430, 1051723530, 4206489449301315, 351597937025844947295000, 73839261438738554611424321993670

Offset: 1

R. H. Hardin May 01 2009

nonn

Cf. A147681.

Column k=8 of A215561.

A147697 Late-growing permutations: number of permutations of 9 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 4862, 22787343150, 807660192541534200, 779279938350147159519336600, 2299118288652572230673921886739695630

Offset: 1

R. H. Hardin, May 01 2009

nonn

Cf. A147681.

Column k=9 of A215561.

A147698 Late-growing permutations: number of permutations of 10 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 16796, 504636071940, 159752979289765273698, 1789294251011628021153241548800, 74651909129454094524656449803220566122458

Offset: 1

R. H. Hardin, May 01 2009

nonn

Cf. A147681.

Column k=10 of A215561.

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.

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A147682 Late-growing permutations: number of permutations of 2 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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A147684 Late-growing permutations: number of permutations of 3 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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Extensions

A147686 Late-growing permutations: number of permutations of 4 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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A147687 Late-growing permutations: number of permutations of 5 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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A147692 Late-growing permutations: number of permutations of 6 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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Extensions

A147694 Late-growing permutations: number of permutations of 7 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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A147695 Late-growing permutations: number of permutations of 8 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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A147697 Late-growing permutations: number of permutations of 9 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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A147698 Late-growing permutations: number of permutations of 10 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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