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%I #15 Feb 27 2017 07:58:42
%S 1,1,2,26,3511,6742796,233249911607,175703195017370516,
%T 3377940832101159287907151,1899957346851645870857879683505890,
%U 35246706696124014829643459097288501560957174,23998872279553738609365779286317516184675391844037227392
%N Number of sequences with j copies of j for each j in {1,2,...,n} and longest increasing subsequence of length n.
%C Sequences counted by a(n) have length A000217(n) and element sum A000330(n).
%H Alois P. Heinz, <a href="/A268667/b268667.txt">Table of n, a(n) for n = 0..15</a>
%H J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. <a href="http://www.ams.org/mathscinet-getitem?mr=681905">MR 681905</a>
%e a(2) = 2: 122, 212.
%e 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.
%p g:= proc(l) option remember; (n-> f(l[1..nops(l)-1])*
%p binomial(n-1, l[-1]-1)+ add(f(sort(subsop(j=l[j]
%p -1, l))), j=1..nops(l)-1))(add(i, i=l))
%p end:
%p f:= l-> (n-> `if`(n<2 or l[-1]=1, 1, `if`(l[1]=0, 0, `if`(
%p n=2, binomial(l[1]+l[2], l[1])-1, g(l)))))(nops(l)):
%p a:= n-> f([$1..n]):
%p seq(a(n), n=0..8);
%t 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]];
%t 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]];
%t a[n_] := f[Range[n]];
%t Table[a[n], {n, 0, 11}] (* _Jean-François Alcover_, Feb 27 2017, after _Alois P. Heinz_ *)
%Y Cf. A000217, A000330, A047909, A268485.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Feb 10 2016
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