A370507
T(n,k) is the number permutations p of [n] that are k-dist-increasing but not j-dist-increasing for any j=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, 7, 11, 0, 1, 9, 22, 33, 55, 0, 1, 19, 77, 112, 192, 319, 0, 1, 34, 189, 480, 788, 1315, 2233, 0, 1, 69, 526, 2187, 3500, 5987, 10409, 17641, 0, 1, 125, 1625, 6811, 18273, 30568, 53791, 92917, 158769, 0, 1, 251, 4111, 23507, 101424, 167480, 299769, 528253, 925337, 1578667
Offset: 0
T(4,1) = 1: 1234.
T(4,2) = 5: 1243, 1324, 2134, 2143, 3142.
T(4,3) = 7: 1342, 1423, 1432, 2314, 2413, 3124, 3214.
T(4,4) = 11: 2341, 2431, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 2, 3;
0, 1, 5, 7, 11;
0, 1, 9, 22, 33, 55;
0, 1, 19, 77, 112, 192, 319;
0, 1, 34, 189, 480, 788, 1315, 2233;
0, 1, 69, 526, 2187, 3500, 5987, 10409, 17641;
0, 1, 125, 1625, 6811, 18273, 30568, 53791, 92917, 158769;
...
-
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:
m:= proc(l) local k;
for k from 0 to nops(l) do if q(l, k)>0 then return k fi od
end:
b:= proc(n) b(n):= add(x^m(l), 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];
m[l_] := Module[{k}, For[k = 0, k <= Length[l], k++, If[q[l, k] > 0, Return[k]]]];
b[n_] := Sum[x^m[l], {l, Permutations[Range@n]}];
T[n_, k_] := Coefficient[b[n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)
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 *)
A370576
a(n) is the difference between the number of n-dist-increasing and (n-1)-dist-increasing permutations p of [2n], where p is k-dist-increasing if k>=0 and p(i)
Original entry on oeis.org
1, 1, 5, 70, 1960, 88200, 5821200, 529729200, 63567504000, 9725828112000, 1847907341280000, 426866595835680000, 117815180450647680000, 38289933646460496000000, 14473594918362067488000000, 6296013789487499357280000000, 3122822839585799681210880000000
Offset: 0
a(2) = 5 = 6 - 1 = |{1234, 1243, 1324, 2134, 2143, 3142}| - |{1234}|.
-
a:= n-> ceil((7/9)*(2*n)!/2^n):
seq(a(n), n=0..22);
# second Maple program:
a:= proc(n) a(n):= `if`(n<4, [1$2, 5, 70][n+1], (2*n-1)*n*a(n-1)) end:
seq(a(n), n=0..22);
Showing 1-3 of 3 results.