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A268699 Total number of sequences with c_j copies of j and longest increasing subsequence of length k summed over all compositions [c_1, c_2, ..., c_k] of n. 4

%I

%S 1,1,2,6,22,95,471,2618,16052,107313,775045,6002106,49536510,

%T 433485429,4004680967,38912323570,396393445096,4221367056961,

%U 46878865762185,541660919690866,6498811587848690,80818650742133717,1040037672241415947,13829246515918840106

%N Total number of sequences with c_j copies of j and longest increasing subsequence of length k summed over all compositions [c_1, c_2, ..., c_k] of n.

%H Alois P. Heinz, <a href="/A268699/b268699.txt">Table of n, a(n) for n = 0..60</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 The compositions of 4 are [1,1,1,1], [2,1,1], [1,2,1], [1,1,2], [2,2], [3,1], [1,3], [4] giving the a(4) = 22 sequences: 1234, 1123, 1213, 1231, 1223, 2123, 1232, 1233, 1323, 3123, 1122, 1212, 1221, 2112, 2121, 1112, 1121, 1211, 1222, 2122, 2212, 1111.

%p c:= l-> f(l)*nops(l)!/(v-> mul(coeff(v, x, j)!,

%p j=0..degree(v)))(add(x^i, i=l)):

%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 h:= (n, i, l)-> `if`(n=0 or i=1, c([1$n, l[]]), h(n, i-1, l)+

%p `if`(i>n, 0, h(n-i, i, [i, l[]]))):

%p a:= n-> h(n$2, []):

%p seq(a(n), n=0..25);

%t c[l_] := f[l]*Length[l]!/Function[v, Product[Coefficient[v, x, j]!, {j, 0, Exponent[v, x]}]][Sum[x^i, {i, l}]];

%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 h[n_, i_, l_] := If[n == 0 || i == 1, c[Join[Array[1&, n], l]], h[n, i-1, l] + If[i > n, 0, h[n-i, i, Join[{i}, l]]]];

%t a[n_] := h[n, n, {}];

%t Table[a[n], {n, 0, 25}] (* _Jean-Fran├žois Alcover_, Sep 06 2022, after _Alois P. Heinz_ *)

%Y Cf. A047909, A268698, A268700, A268701.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Feb 11 2016

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Last modified October 4 21:22 EDT 2022. Contains 357240 sequences. (Running on oeis4.)