cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A321273 Sum over all permutations of [n] of the maximum of the lengths of increasing or decreasing subsequences.

Original entry on oeis.org

1, 4, 14, 70, 396, 2628, 20270, 175392, 1686374, 17920528, 208454628, 2629931688, 35774761662, 522351495684, 8149929922408, 135284126840592, 2380119357533974, 44243729657494640, 866599471539160876, 17839886344238238784, 385065445154671172880, 8695565142604747421416
Offset: 1

Views

Author

Alois P. Heinz, Nov 01 2018

Keywords

Links

Crossrefs

Cf. A003316, A321274, A321275, A321276, A321277, A321278.

Programs

  • Maple
    h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
    l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*max(l[1], nops(l)):
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
     g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);
  • Mathematica
    h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[j > l[[k]], 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]][Length[l]];
f[l_] := h[l]^2 Max[l[[1]], Length[l]];
g[n_, i_, l_] := If[n == 0 || i == 1, f[Join[l, Table[1, {n}]]], g[n, i - 1, l] + g[n - i, Min[i, n - i], Append[l, i]]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Oct 31 2021, after Alois P. Heinz *)

Formula

A321274(n) < A003316(n) < a(n) for n > 1.

A321274 Sum over all permutations of [n] of the minimum of the lengths of longest increasing subsequence and longest decreasing subsequence.

Original entry on oeis.org

1, 2, 10, 46, 274, 1894, 14660, 128648, 1259740, 13540882, 158689006, 2018664332, 27699652406, 407457326286, 6395402111042, 106731605965344, 1887716456363316, 35269257369001618, 694027051724655398, 14346767204627002964, 310852440258761877068, 7045172291061429434354
Offset: 1

Views

Author

Alois P. Heinz, Nov 01 2018

Keywords

Links

Crossrefs

Cf. A003316, A321273, A321275, A321276, A321277, A321278.

Programs

  • Maple
    h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
    l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*min(l[1], nops(l)):
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
     g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);

Formula

a(n) < A003316(n) < A321273(n) for n > 1.

A321277 One half of the sum over all permutations of [n] of the absolute difference between the length of the longest increasing subsequence and the length of the longest decreasing subsequence.

Original entry on oeis.org

0, 1, 2, 12, 61, 367, 2805, 23372, 213317, 2189823, 24882811, 305633678, 4037554628, 57447084699, 877263905683, 14276260437624, 246201450585329, 4487236144246511, 86286209907252739, 1746559569805617910, 37106502447954647906, 825196425771658993531
Offset: 1

Views

Author

Alois P. Heinz, Nov 01 2018

Keywords

Links

Crossrefs

Cf. A003316, A321273, A321274, A321275, A321276, A321278, A321316.

Programs

  • Maple
    h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
    l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*abs(l[1]-nops(l))/2:
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
     g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);

Formula

a(n) = (1/2) * Sum_{k=1-n..n-1} abs(k) * A321316(n,k).

A321316 Number T(n,k) of permutations of [n] whose difference between the length of the longest increasing subsequence and the length of the longest decreasing subsequence equals k; triangle T(n,k), n >= 1, 1-n <= k <= n-1, read by rows.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 4, 0, 1, 1, 0, 9, 4, 9, 0, 1, 1, 0, 16, 25, 36, 25, 16, 0, 1, 1, 0, 25, 81, 125, 256, 125, 81, 25, 0, 1, 1, 0, 36, 196, 421, 1225, 1282, 1225, 421, 196, 36, 0, 1, 1, 0, 49, 400, 1225, 4292, 9261, 9864, 9261, 4292, 1225, 400, 49, 0, 1
Offset: 1

Views

Author

Alois P. Heinz, Nov 03 2018

Keywords

Examples

			:                                1                             ;
:                          1,    0,    1                       ;
:                    1,    0,    4,    0,   1                  ;
:               1,   0,    9,    4,    9,   0,   1             ;
:          1,   0,  16,   25,   36,   25,  16,   0,  1         ;
:      1,  0,  25,  81,  125,  256,  125,  81,  25,  0, 1      ;
:   1, 0, 36, 196, 421, 1225, 1282, 1225, 421, 196, 36, 0, 1   ;

Links

Crossrefs

Column k=0 gives A321313.
Row sums give A000142.
T(n+1,n-2) gives A000290.
Cf. A303697, A321277, A321278, A321314, A321315.

Programs

  • Maple
    h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
    l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*x^(l[1]-nops(l)) :
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
     g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
b:= proc(n) option remember; g(n$2, []) end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=1-n..n-1), n=1..10);
  • Mathematica
    h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[j > l[[k]], 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
f[l_] := h[l]^2*x^(l[[1]] - Length[l]);
g[n_, i_, l_] := If[n == 0 || i == 1, f[Join[l, Table[1, {n}]]], g[n, i - 1, l] + g[n - i, Min[i, n - i], Append[l, i]]];
b[n_] := b[n] = g[n, n, {}];
T[n_, k_] := Coefficient[b[n], x, k];
Table[Table[T[n, k], {k, 1 - n, n - 1}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Feb 27 2021, after Alois P. Heinz *)

Formula

T(n,k) = T(n,-k).
Sum_{k=1..n-1} T(n,k) = A321314(n).
Sum_{k=0..n-1} T(n,k) = A321315(n).
(1/2) * Sum_{k=1-n..n-1} abs(k) * T(n,k) = A321277(n).
(1/2) * Sum_{k=1-n..n-1} k^2 * T(n,k) = A321278(n).

A321275 Sum over all permutations of [n] of the product of the lengths of longest increasing subsequence and longest decreasing subsequence.

Original entry on oeis.org

1, 4, 22, 132, 890, 6812, 58422, 555900, 5819658, 66554180, 825839718, 11054124886, 158795559000, 2437248222710, 39809464449676, 689538524084168, 12625142440334342, 243656361772961292, 4943801229819987022, 105212500452414418118, 2343513475564027153128
Offset: 1

Views

Author

Alois P. Heinz, Nov 01 2018

Keywords

Links

Crossrefs

Cf. A003316, A321273, A321274, A321276, A321277, A321278.

Programs

  • Maple
    h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
    l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*l[1]*nops(l):
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
     g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);

A321276 Sum over all permutations of [n] of the length of the longest increasing subsequence raised to the power of the length of the longest decreasing subsequence.

Original entry on oeis.org

1, 3, 20, 174, 1915, 25861, 407691, 7330188, 148016449, 3312032213, 81207824255, 2162810487154, 62125097028962, 1913156511113517, 62839800627095263, 2191735865280260976, 80859575674731497805, 3144804693463679033629, 128550453029684197431607
Offset: 1

Views

Author

Alois P. Heinz, Nov 01 2018

Keywords

Links

Crossrefs

Cf. A003316, A321273, A321274, A321275, A321277, A321278.

Programs

  • Maple
    h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
    l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*l[1]^nops(l):
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
     g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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