A047909 -id:A047909 - cp's OEIS frontend

A268850 Number of sequences with 7 copies each of 1,2,...,n and longest increasing subsequence of length n.

1, 1, 3431, 397222288, 460827731023773, 2931247600219365331976, 70803267480031877368227941803, 5078529731893937404909347067888886466, 909546798992441266072332791609067485208949369, 358281333933096129012031117609647623312585201668494007

Offset: 0

Alois P. Heinz, Feb 14 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..80
J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Row n=7 of A047909.

Table[Sum[Sum[Sum[Sum[Sum[Sum[k!/(i1!*i2!*i3!*i4!*i5!*i6!*(k - i1 - i2 - i3 - i4 - i5 - i6)!)*(7*k)!/(i1 + 2*i2 + 3*i3 + 4*i4 + 5*i5 + 6*i6 + 7*(k - i1 - i2 - i3 - i4 - i5 - i6))!*(-1)^(i1 + 2*i2 + 3*i3 + 4*i4 + 5*i5 + 6*i6 + 7*(k - i1 - i2 - i3 - i4 - i5 - i6) - k)/(720^i1*120^i2*24^i3*6^i4*2^i5), {i6, 0, k - i1 - i2 - i3 - i4 - i5}], {i5, 0, k - i1 - i2 - i3 - i4}], {i4, 0, k - i1 - i2 - i3}], {i3, 0, k - i1 - i2}], {i2, 0, k - i1}], {i1, 0, k}], {k, 0, 10}] (* Vaclav Kotesovec, Mar 02 2016, after Horton and Kurn *)

a(n) ~ sqrt(7) * (7^7/6!)^n * n^(6*n) / exp(6*(n+1)). - Vaclav Kotesovec, Mar 03 2016

A268851 Number of sequences with 8 copies each of 1,2,...,n and longest increasing subsequence of length n.

Original entry on oeis.org

1, 1, 12869, 9450343019, 98540942707986273, 7370846583668954571029069, 2612508237897293571677286548812861, 3315159778348807570604149155371730111763599, 12324197596430667064913735085330208112438377122058241

Offset: 0

Alois P. Heinz, Feb 14 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..70
J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Row n=8 of A047909.

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k!/(i1!*i2!*i3!*i4!*i5!*i6!*i7!*(k - i1 - i2 - i3 - i4 - i5 - i6 - i7)!)*(8*k)!/(i1 + 2*i2 + 3*i3 + 4*i4 + 5*i5 + 6*i6 + 7*i7 + 8*(k - i1 - i2 - i3 - i4 - i5 - i6 - i7))!*(-1)^(i1 + 2*i2 + 3*i3 + 4*i4 + 5*i5 + 6*i6 + 7*i7 + 8*(k - i1 - i2 - i3 - i4 - i5 - i6 - i7) - k)/(5040^i1 * 720^i2 * 120^i3 * 24^i4 * 6^i5 * 2^i6), {i7, 0, k - i1 - i2 - i3 - i4 - i5 - i6}], {i6, 0, k - i1 - i2 - i3 - i4 - i5}], {i5, 0, k - i1 - i2 - i3 - i4}], {i4, 0, k - i1 - i2 - i3}], {i3, 0, k - i1 - i2}], {i2, 0, k - i1}], {i1, 0, k}], {k, 0, 10}] (* Vaclav Kotesovec, Mar 02 2016, after Horton and Kurn *)

a(n) ~ sqrt(8) * (8^8/7!)^n * n^(7*n) / exp(7*(n+1)). - Vaclav Kotesovec, Mar 03 2016

A268852 Number of sequences with 9 copies each of 1,2,...,n and longest increasing subsequence of length n.

Original entry on oeis.org

1, 1, 48619, 227749730869, 21364658238692907265, 18683332440278067962764855531, 96042041352156959435669839199503441435, 2124172213523649116114190361767338538457819064671, 161347197004751609388708454579308609212572710243373701247489

Offset: 0

Alois P. Heinz, Feb 14 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..65
J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Row n=9 of A047909.

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k!/(i1!*i2!*i3!*i4!*i5!*i6!*i7!*i8!*(k - i1 - i2 - i3 - i4 - i5 - i6 - i7 - i8)!)*(9*k)!/(i1 + 2*i2 + 3*i3 + 4*i4 + 5*i5 + 6*i6 + 7*i7 + 8*i8 + 9*(k - i1 - i2 - i3 - i4 - i5 - i6 - i7 - i8))!*(-1)^(i1 + 2*i2 + 3*i3 + 4*i4 + 5*i5 + 6*i6 + 7*i7 + 8*i8 + 9*(k - i1 - i2 - i3 - i4 - i5 - i6 - i7 - i8) - k)/(8!^i1 * 7!^i2 * 6!^i3 * 5!^i4 * 4!^i5 * 3!^i6 * 2!^i7), {i8, 0, k - i1 - i2 - i3 - i4 - i5 - i6 - i7}], {i7, 0, k - i1 - i2 - i3 - i4 - i5 - i6}], {i6, 0, k - i1 - i2 - i3 - i4 - i5}], {i5, 0, k - i1 - i2 - i3 - i4}], {i4, 0, k - i1 - i2 - i3}], {i3, 0, k - i1 - i2}], {i2, 0, k - i1}], {i1, 0, k}], {k, 0, 10}] (* Vaclav Kotesovec, Mar 02 2016, after Horton and Kurn *)

a(n) ~ 3 * (9^9/8!)^n * n^(8*n) / exp(8*(n+1)). - Vaclav Kotesovec, Mar 03 2016

A268853 Number of sequences with 10 copies each of 1,2,...,n and longest increasing subsequence of length n.

Original entry on oeis.org

1, 1, 184755, 5549991941777, 4697818999010952011441, 47964531978782851644184417448714, 3553102771891168237056005934820411063204249, 1355554085495648757684163048897568469564674091083870680, 2077847308887546704733072843165544143697549966176523511722695300153

Offset: 0

Alois P. Heinz, Feb 14 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..60
J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Row n=10 of A047909.

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k!/(i1!*i2!*i3!*i4!*i5!*i6!* i7!*i8!*i9!*(k - i1 - i2 - i3 - i4 - i5 - i6 - i7 - i8 - i9)!)*(10*k)!/(i1 + 2*i2 + 3*i3 + 4*i4 + 5*i5 + 6*i6 + 7*i7 + 8*i8 + 9*i9 + 10*(k - i1 - i2 - i3 - i4 - i5 - i6 - i7 - i8 - i9))!*(-1)^(i1 + 2*i2 + 3*i3 + 4*i4 + 5*i5 + 6*i6 + 7*i7 + 8*i8 + 9*i9 + 10*(k - i1 - i2 - i3 - i4 - i5 - i6 - i7 - i8 - i9) - k)/(9!^i1 * 8!^i2 * 7!^i3 * 6!^i4 * 5!^i5 * 4!^i6 * 3!^i7 * 2!^i8), {i9, 0, k - i1 - i2 - i3 - i4 - i5 - i6 - i7 - i8}], {i8, 0, k - i1 - i2 - i3 - i4 - i5 - i6 - i7}], {i7, 0, k - i1 - i2 - i3 - i4 - i5 - i6}], {i6, 0, k - i1 - i2 - i3 - i4 - i5}], {i5, 0, k - i1 - i2 - i3 - i4}], {i4, 0, k - i1 - i2 - i3}], {i3, 0, k - i1 - i2}], {i2, 0, k - i1}], {i1, 0, k}], {k, 0, 10}] (* Vaclav Kotesovec, Mar 02 2016, after Horton and Kurn *)

a(n) ~ sqrt(10) * (10^10/9!)^n * n^(9*n) / exp(9*(n+1)). - Vaclav Kotesovec, Mar 03 2016

A268667 Number of sequences with j copies of j for each j in {1,2,...,n} and longest increasing subsequence of length n.

Original entry on oeis.org

1, 1, 2, 26, 3511, 6742796, 233249911607, 175703195017370516, 3377940832101159287907151, 1899957346851645870857879683505890, 35246706696124014829643459097288501560957174, 23998872279553738609365779286317516184675391844037227392

Offset: 0

Alois P. Heinz, Feb 10 2016

nonn

Sequences counted by a(n) have length A000217(n) and element sum A000330(n).

			a(2) = 2: 122, 212.
a(3) = 26: 122333, 123233, 123323, 123332, 132233, 132323, 132332, 133223, 133232, 212333, 213233, 213323, 231233, 231323, 233123, 312233, 312323, 312332, 313223, 313232, 321233, 321323, 323123, 331223, 331232, 332123.

Alois P. Heinz, Table of n, a(n) for n = 0..15
J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Cf. A000217, A000330, A047909, A268485.

g:= proc(l) option remember; (n-> f(l[1..nops(l)-1])*
      binomial(n-1, l[-1]-1)+ add(f(sort(subsop(j=l[j]
      -1, l))), j=1..nops(l)-1))(add(i, i=l))
    end:
f:= l-> (n-> `if`(n<2 or l[-1]=1, 1, `if`(l[1]=0, 0, `if`(
         n=2, binomial(l[1]+l[2], l[1])-1, g(l)))))(nops(l)):
a:= n-> f([$1..n]):
seq(a(n), n=0..8);

g[l_] := g[l] = Function[n, f[Most[l]]*Binomial[n-1, l[[-1]]-1] + Sum[f[ Sort[ ReplacePart[l, j -> l[[j]]-1]]], {j, 1, Length[l]-1}]][Total[l]];
f[l_] := Function[n, If[n<2 || l[[-1]]==1, 1, If[l[[1]]==0, 0, If[n==2, Binomial[l[[1]] + l[[2]], l[[1]]]-1, g[l]]]]][Length[l]];
a[n_] := f[Range[n]];
Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Feb 27 2017, after Alois P. Heinz *)

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A268850 Number of sequences with 7 copies each of 1,2,...,n and longest increasing subsequence of length n.

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A268851 Number of sequences with 8 copies each of 1,2,...,n and longest increasing subsequence of length n.

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A268852 Number of sequences with 9 copies each of 1,2,...,n and longest increasing subsequence of length n.

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A268853 Number of sequences with 10 copies each of 1,2,...,n and longest increasing subsequence of length n.

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Mathematica

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A268667 Number of sequences with j copies of j for each j in {1,2,...,n} and longest increasing subsequence of length n.

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