A262495
Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order such that sorts of adjacent parts are different; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 4, 1, 0, 1, 6, 12, 7, 1, 0, 1, 10, 31, 33, 11, 1, 0, 1, 14, 73, 130, 77, 16, 1, 0, 1, 21, 165, 464, 438, 157, 22, 1, 0, 1, 29, 357, 1558, 2216, 1223, 289, 29, 1, 0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1
Offset: 0
T(3,1) = 1: 3a.
T(3,2) = 2: 2a1b, 1a1b1a.
T(3,3) = 1: 1a1b1c.
T(5,3) = 12: 3a1b1c, 2a2b1c, 2a1b1a1c, 2a1b1c1a, 2a1b1c1b, 1a1b1a1b1c, 1a1b1a1c1a, 1a1b1a1c1b, 1a1b1c1a1b, 1a1b1c1a1c, 1a1b1c1b1a, 1a1b1c1b1c.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 2, 1;
0, 1, 4, 4, 1;
0, 1, 6, 12, 7, 1;
0, 1, 10, 31, 33, 11, 1;
0, 1, 14, 73, 130, 77, 16, 1;
0, 1, 21, 165, 464, 438, 157, 22, 1;
0, 1, 29, 357, 1558, 2216, 1223, 289, 29, 1;
0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1;
...
Columns k=0-10 give:
A000007,
A057427,
A000065,
A262445/6 =
A320545,
A320546,
A320547,
A320548,
A320549,
A320550,
A320551,
A320552.
-
b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
T:= (n, k)-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n-1), b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]; A[n_, k_] := If[n == 0, 1, If[k<2, k, k*b[n, n, k-1]]]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2017, translated from Maple *)
A258458
Number of partitions of n into parts of exactly 3 sorts which are introduced in ascending order.
Original entry on oeis.org
1, 7, 33, 130, 463, 1557, 5031, 15877, 49240, 151116, 460173, 1394645, 4212071, 12693724, 38195286, 114817389, 344911117, 1035659955, 3108817911, 9330152740, 27997803871, 84008165515, 252053831034, 756220333901, 2268778132337, 6806569134920, 20420175154486
Offset: 3
-
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,3):
seq(a(n), n=3..35);
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i/(i!*(k - i)!), {i, 0, k}];
a[n_] := T[n, 3];
Table[a[n], {n, 3, 35}] (* Jean-François Alcover, May 22 2018, translated from Maple *)
A262445
Number of exact 3-colored partitions such that no adjacent parts have the same color.
Original entry on oeis.org
0, 0, 0, 6, 24, 72, 186, 438, 990, 2142, 4560, 9492, 19620, 40068, 81534, 164892, 332808, 669528, 1345554, 2699448, 5412636, 10843038, 21714972, 43467342, 86995428, 174069306, 348265164, 696694692, 1393652298, 2787646380, 5575837836, 11152384044, 22305891948, 44613248352, 89228806704, 178460625402, 356925987924
Offset: 0
a(3)=6 because there are three partitions of 3 and there are no ways to color [3] or [2,1] but there are six ways to color [1,1,1].
-
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
end:
a:= n-> `if`(n=0, 0, b(n$2, 2)/2*3-6*b(n$2, 1)+3):
seq(a(n), n=0..40); # Alois P. Heinz, Sep 23 2015
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]]; a[n_] := If[n == 0, 0, b[n, n, 2]/2*3 - 6*b[n, n, 1] + 3]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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