A268698
Total number of sequences with p_j copies of j and longest increasing subsequence of length k summed over all partitions [p_1, p_2, ..., p_k] of n.
Original entry on oeis.org
1, 1, 2, 4, 13, 36, 157, 554, 2800, 12530, 70772, 362188, 2370564, 13658713, 95366064, 642861687, 4774830263, 34769374156, 288999332899, 2255537559077, 19693313843687, 172690825379198, 1572921138465599, 14538979953843188, 145980379536597239
Offset: 0
The partitions of 4 are [1,1,1,1], [2,1,1], [2,2], [3,1], [4] giving the a(4) = 13 sequences: 1234, 1123, 1213, 1231, 1122, 1212, 1221, 2112, 2121, 1112, 1121, 1211, 1111.
-
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)):
h:= (n, i, l)-> `if`(n=0 or i=1, f([1$n, l[]]), h(n, i-1, l)+
`if`(i>n, 0, h(n-i, i, [i, l[]]))):
a:= n-> h(n$2, []):
seq(a(n), n=0..25);
-
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]]; h[n_, i_, l_] := h[n, i, l] = If[n == 0 || i == 1, f[Join[Array[1&, n], l]], h[n, i - 1, l] + If[i>n, 0, h[n-i, i, Prepend[l, i]]]]; a[n_] := h[n, n, {}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)
A268700
Total number of sequences with p_j copies of j and longest increasing subsequence of length k summed over all partitions [p_1, p_2, ..., p_k] of n into distinct parts.
Original entry on oeis.org
1, 1, 1, 3, 4, 14, 46, 111, 330, 1614, 7348, 21340, 98145, 379405, 2633085, 14871033, 57284558, 278927415, 1609313975, 8289565670, 74945364815, 522977754235, 2403799401259, 14180489136597, 83964652635668, 623008803758260, 3918144764978718, 46950727351392315
Offset: 0
The partitions of 4 into distinct parts are [3,1], [4] giving the a(4) = 4 sequences: 1112, 1121, 1211, 1111.
-
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)):
h:= (n, i, l)-> `if`(n>i*(i+1)/2, 0, `if`(n=0, f(l), h(n, i-1, l)
+`if`(i>n, 0, h(n-i, i-1, [i, l[]])))):
a:= n-> h(n$2, []):
seq(a(n), n=0..30);
-
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]]; h[n_, i_, l_] := If[n>i*(i+1)/2, 0, If[n==0, f[l], h[n, i-1, l] + If[i>n, 0, h[n-i, i-1, Join[{i}, l]]]]]; a[n_] := h[n, n, {}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 11 2017, translated from Maple *)
A268701
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 into distinct parts.
Original entry on oeis.org
1, 1, 1, 5, 7, 27, 195, 421, 1619, 8675, 105757, 274029, 1402193, 6625987, 55349787, 975068069, 3137395939, 17960895375, 101880880461, 696011551909, 7596647200175, 197122787505191, 723879298052695, 4905597865756059, 29537689035766501, 227793692735075911
Offset: 0
The compositions of 4 into distinct parts are [3,1], [1,3], [4] giving the a(4) = 7 sequences: 1112, 1121, 1211, 1222, 2122, 2212, 1111.
-
c:= l-> f(l)*nops(l)!/(v-> mul(coeff(v, x, j)!,
j=0..degree(v)))(add(x^i, i=l)):
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)):
h:= (n, i, l)-> `if`(n>i*(i+1)/2, 0, `if`(n=0, c(l), h(n, i-1, l)
+`if`(i>n, 0, h(n-i, i-1, [i, l[]])))):
a:= n-> h(n$2, []):
seq(a(n), n=0..30);
-
c[l_] := f[l]*Length[l]!/Function[v, Product[Coefficient[v, x, j]!,{j, 0, Exponent[v, x]}]][Sum[x^i, {i, l}]];
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]];
h[n_, i_, l_] := If[n > i (i + 1)/2, 0, If[n == 0, c[l], h[n, i - 1, l] + If[i > n, 0, h[n - i, i - 1, Join[{i}, l]]]]];
a[n_] := h[n, n, {}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)
Showing 1-3 of 3 results.