A248686
Triangular array of multinomial coefficients: T(n,k) = n!/(n(1)!*n(2)!* ... *n(k)!), where n(i) = floor((n + i - 1)/k) for i = 1 .. k.
Original entry on oeis.org
1, 1, 2, 1, 3, 6, 1, 6, 12, 24, 1, 10, 30, 60, 120, 1, 20, 90, 180, 360, 720, 1, 35, 210, 630, 1260, 2520, 5040, 1, 70, 560, 2520, 5040, 10080, 20160, 40320, 1, 126, 1680, 7560, 22680, 45360, 90720, 181440, 362880, 1, 252, 4200, 25200, 113400, 226800, 453600, 907200, 1814400, 3628800
Offset: 1
First seven rows:
1
1 2
1 3 6
1 6 12 24
1 10 30 60 120
1 20 90 180 360 720
1 35 210 630 1260 2520 5040
...
Writing floor as [ ], the numbers comprising row 4 are
T(4,1) = 4!/[4/1]! = 24/24 = 1
T(4,2) = 4!/([4/2]![5/2]!) = 24/(2*2) = 6
T(4,3) = 4!/([4/3]![5/3]![6/3]!) = 24/(1*1*2) = 12
T(4,4) = 4!/([4/4]![5/4]![6/4]![7/4]!) = 24/(1*1*1*1) = 24.
-
T:= (n, k)-> combinat[multinomial](n, floor((n+i)/k)$i=0..k-1):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Feb 09 2023
-
f[n_, k_] := f[n, k] = n!/Product[Floor[(n + i)/k]!, {i, 0, k - 1}]
t = Table[f[n, k], {n, 0, 10}, {k, 1, n}];
u = Flatten[t] (* A248686 sequence *)
TableForm[t] (* A248686 array *)
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 22}] (* A248687 *)
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 *)
Showing 1-3 of 3 results.
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