A317446 -id:A317446 - cp's OEIS frontend

A317130 Number of permutations of [n] whose lengths of increasing runs are triangular numbers.

1, 1, 1, 2, 7, 24, 93, 483, 2832, 17515, 123226, 978405, 8312802, 75966887, 756376739, 8070649675, 91320842018, 1099612368110, 14054043139523, 189320856378432, 2682416347625463, 39945105092501742, 623240458310527252, 10160826473676346731, 172871969109661492526

Offset: 0

Alois P. Heinz, Jul 21 2018

nonn

			a(2) = 1: 21.
a(3) = 2: 123, 321.
a(4) = 7: 1243, 1342, 2134, 2341, 3124, 4123, 4321.
a(5) = 24: 12543, 13542, 14532, 21354, 21453, 23541, 24531, 31254, 31452, 32145, 32451, 34521, 41253, 41352, 42135, 42351, 43125, 51243, 51342, 52134, 52341, 53124, 54123, 54321.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A000217, A097597, A193374, A317111, A317128, A317129, A317131, A317132, A317446.

g:= n-> `if`(issqr(8*n+1), 1, 0):
b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),
      `if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+
       add(b(u+j-1, o-j, t+1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..27);

g[n_] := If[IntegerQ @ Sqrt[8n+1], 1, 0];
b[u_, o_, t_] := b[u, o, t] = If[u+o==0, g[t], If[g[t]==1, Sum[b[u-j, o+j-1, 1], {j, 1, u}], 0] + Sum[b[u+j-1, o-j, t+1], {j, 1, o}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 27] (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)

A317444 Number of permutations of [n] whose lengths of increasing runs are distinct Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 5, 6, 19, 212, 40, 757, 2170, 13546, 379084, 8978, 73195, 2702092, 772852, 38833826, 213557110, 2390871412, 150689939006, 9394670, 634504029, 4522073096, 63395566566, 5160905755362, 192831696582, 3068824154606, 289158899744046, 116561588867106

Offset: 0

Alois P. Heinz, Jul 28 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A000045, A317128, A317445, A317446, A317447, A317448.

g:= (n, s)-> `if`(n in s or not
    (issqr(5*n^2+4) or issqr(5*n^2-4)), 0, 1):
b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
      `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
       , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
    end:
a:= n-> b(n, 0$2, {}):
seq(a(n), n=0..30);

g[n_, s_] := If[MemberQ[s, n] || !(
     IntegerQ@Sqrt[5*n^2 + 4] || IntegerQ@Sqrt[5*n^2 - 4]), 0, 1];
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
     If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
     {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
a[n_] := b[n, 0, 0, {}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)

A317445 Number of permutations of [n] whose lengths of increasing runs are distinct squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 8, 0, 0, 0, 1, 18, 0, 0, 1428, 47998, 0, 1, 32, 0, 0, 9688, 505056, 0, 0, 0, 4085949, 284958912, 0, 0, 290824632172, 28643427712626, 0, 0, 0, 104902510, 9998016202, 1, 72, 23207824626842, 3008268832634364, 182778, 206173972520, 24290829974718, 0

Offset: 0

Alois P. Heinz, Jul 28 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000290, A001422, A003995, A317129, A317444, A317446, A317447, A317448.

g:= (n, s)-> `if`(n in s or not issqr(n), 0, 1):
b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
      `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
       , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
    end:
a:= n-> b(n, 0$2, {}):
seq(a(n), n=0..50);

g[n_, s_] := If[MemberQ[s, n] || !IntegerQ@Sqrt[n], 0, 1];
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
     If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
     {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
a[n_] := b[n, 0, 0, {}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 24 2021, after Alois P. Heinz *)

a(n) = 0 <=> n in { A001422 }.

a(n) > 0 <=> n in { A003995 }.

A317447 Number of permutations of [n] whose lengths of increasing runs are distinct prime numbers.

Original entry on oeis.org

1, 0, 1, 1, 0, 19, 0, 41, 110, 70, 13696, 1, 44796, 155, 411064, 2122802, 251746, 1057634441, 4404368, 25043183, 44848672, 19725545894, 106293316, 307873058001, 50194102, 8305023165502, 65808841818130, 33715371370134, 115625740201672616, 78940089764191

Offset: 0

Alois P. Heinz, Jul 28 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A000040, A317131, A317444, A317445, A317446, A317448.

g:= (n, s)-> `if`(n in s or not (n=0 or isprime(n)), 0, 1):
b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
      `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
       , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
    end:
a:= n-> b(n, 0$2, {}):
seq(a(n), n=0..40);

g[n_, s_] := If[MemberQ[s, n] || Not [n == 0 || PrimeQ[n]], 0, 1];
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
     If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
     {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
a[n_] := b[n, 0, 0, {}];
a /@ Range[0, 40] (* Jean-François Alcover, Mar 29 2021, after Alois P. Heinz *)

from functools import lru_cache
from sympy import isprime
def g(n, s): return int((n == 0 or isprime(n)) and not n in s)
@lru_cache(maxsize=None)
def b(u, o, t, s):
  if u + o == 0: return g(t, s)
  c1 = sum(b(u-j, o+j-1, 1, tuple(sorted(s+(t,)))) for j in range(1, u+1)) if g(t, s) else 0
  return c1 + sum(b(u+j-1, o-j, t+1, s) for j in range(1, o+1))
def a(n): return b(n, 0, 0, tuple())
print([a(n) for n in range(41)]) # Michael S. Branicky, Mar 29 2021 after Alois P. Heinz

A317448 Number of permutations of [n] whose lengths of increasing runs are distinct factorial numbers.

Original entry on oeis.org

1, 1, 1, 4, 0, 0, 1, 12, 54, 1002, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 48, 648, 39444, 0, 0, 1187548, 96978608, 1721374454, 169149221140, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Alois P. Heinz, Jul 28 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..721

Cf. A000142, A059590, A115945, A317132, A317444, A317445, A317446, A317447.

h:= proc(n) local i; 1; for i from 2 do
      if n=% then 1; break elif n<% then 0; break fi;
      %*i od; h(n):=%
    end:
g:= (n, s)-> `if`(n in s or not (n=0 or h(n)=1), 0, 1):
b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
      `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
       , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
    end:
a:= n-> b(n, 0$2, {}):
seq(a(n), n=0..34);

h[n_] := Module[{i, pc = 1}, For[i = 2, True, i++, Which[n == pc, pc = 1; Break[], n < pc, pc = 0; Break[]]; pc = pc*i]; h[n] = pc];
g[n_, s_] := If[MemberQ[s, n] || !(n == 0 || h[n] == 1), 0, 1];
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
     If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
     {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
a[n_] := b[n, 0, 0, {}];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)

a(n) = 0 <=> n in { A115945 }.

a(n) > 0 <=> n in { A059590 }.

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A317130 Number of permutations of [n] whose lengths of increasing runs are triangular numbers.

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A317444 Number of permutations of [n] whose lengths of increasing runs are distinct Fibonacci numbers.

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A317445 Number of permutations of [n] whose lengths of increasing runs are distinct squares.

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A317447 Number of permutations of [n] whose lengths of increasing runs are distinct prime numbers.

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A317448 Number of permutations of [n] whose lengths of increasing runs are distinct factorial numbers.

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