A370505
T(n,k) is the difference between the number of k-dist-increasing and (k-1)-dist-increasing permutations of [n], where p is k-dist-increasing if k>=0 and p(i)=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 5, 6, 12, 0, 1, 9, 20, 30, 60, 0, 1, 19, 70, 90, 180, 360, 0, 1, 34, 175, 420, 630, 1260, 2520, 0, 1, 69, 490, 1960, 2520, 5040, 10080, 20160, 0, 1, 125, 1554, 5880, 15120, 22680, 45360, 90720, 181440, 0, 1, 251, 3948, 21000, 88200, 113400, 226800, 453600, 907200, 1814400
Offset: 0
T(0,0) = 1: (only) the empty permutation is 0-dist-increasing.
T(4,2) = 5 = 6 - 1 = |{1234, 1243, 1324, 2134, 2143, 3142}| - |{1234}|.
Permutation 3142 is 2-dist-increasing and 4-dist-increasing but not 3-dist-increasing.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 2, 3;
0, 1, 5, 6, 12;
0, 1, 9, 20, 30, 60;
0, 1, 19, 70, 90, 180, 360;
0, 1, 34, 175, 420, 630, 1260, 2520;
0, 1, 69, 490, 1960, 2520, 5040, 10080, 20160;
0, 1, 125, 1554, 5880, 15120, 22680, 45360, 90720, 181440;
...
-
b:= proc(n, k) option remember; `if`(k<1,
`if`(n=k, 1, 0), n!/mul(iquo(n+i, k)!, i=0..k-1))
end:
T:= (n, k)-> b(n, k)-b(n, k-1):
seq(seq(T(n, k), k=0..n), n=0..10);
A370506
T(n,k) is the number permutations p of [n] that are j-dist-increasing for exactly k distinct values j in [n], where p is j-dist-increasing if j>=0 and p(i)=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 11, 8, 4, 1, 0, 55, 38, 19, 7, 1, 0, 319, 228, 110, 50, 12, 1, 0, 2233, 1574, 775, 322, 115, 20, 1, 0, 17641, 12524, 6216, 2611, 1033, 261, 33, 1, 0, 158769, 112084, 55692, 23585, 9103, 3006, 586, 54, 1, 0, 1578667, 1119496, 556754, 238425, 91764, 33929, 8372, 1304, 88, 1
Offset: 0
T(4,1) = 11: 2341, 2431, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321.
T(4,2) = 8: 1342, 1423, 1432, 2314, 2413, 3124, 3142, 3214.
T(4,3) = 4: 1243, 1324, 2134, 2143.
T(4,4) = 1: 1234.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 3, 2, 1;
0, 11, 8, 4, 1;
0, 55, 38, 19, 7, 1;
0, 319, 228, 110, 50, 12, 1;
0, 2233, 1574, 775, 322, 115, 20, 1;
0, 17641, 12524, 6216, 2611, 1033, 261, 33, 1;
0, 158769, 112084, 55692, 23585, 9103, 3006, 586, 54, 1;
...
-
q:= proc(l, k) local i; for i from 1 to nops(l)-k do
if l[i]>=l[i+k] then return 0 fi od; 1
end:
b:= proc(n) option remember; add(x^add(
q(l, j), j=1..n), l=combinat[permute](n))
end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n,k), k=0..n), n=0..8);
-
q[l_, k_] := Module[{i}, For[i = 1, i <= Length[l]-k, i++,
If[l[[i]] >= l[[i+k]], Return@0]]; 1];
b[n_] := b[n] = Sum[x^Sum[q[l, j], {j, 1, n}], {l, Permutations[Range[n]]}];
T[n_, k_] := Coefficient[b[n], x, k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 24 2024, after Alois P. Heinz *)
A376455
a(n) = least k such that n^(2k+1)/(2k+1)! < 1.
Original entry on oeis.org
1, 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 83, 84, 85, 87, 88
Offset: 0
Cf.
A370507,
A376284,
A376952,
A376953,
A376954,
A376955,
A376956,
A376957,
A376958,
A376959,
A376960.
-
a[n_] := Select[Range[z], n^(2 # + 1)/(2 # + 1)! < 1 &, 1]
Flatten[Table[a[n], {n, 0, 100}]]
A370514
Number of permutations p of [n] such that for each distance d in [n-1] there is at least one index i in [n-d] with p(i)>p(i+d).
Original entry on oeis.org
1, 1, 1, 3, 11, 55, 319, 2233, 17641, 158769, 1578667, 17365337, 207865289, 2702248757, 37786779669, 566801695035, 9063808803203, 154084749654451
Offset: 0
a(0) = 1: the empty permutation.
a(1) = 1: 1.
a(2) = 1: 21.
a(3) = 3: 231, 312, 321.
a(4) = 11: 2341, 2431, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321.
a(5) = 55: 23451, 23541, 24351, 24531, ..., 54213, 54231, 54312, 54321.
a(6) = 319: 234561, 234651, 235461, 235641, ..., 654213, 654231, 654312, 654321.
Comments