A356263
Triangle read by rows. The reduced triangle of the partition triangle of irreducible permutations (A356262). T(n, k) for n >= 1 and 0 <= k < n.
Original entry on oeis.org
1, 0, 1, 0, 2, 1, 0, 3, 9, 1, 0, 5, 41, 24, 1, 0, 8, 150, 247, 55, 1, 0, 14, 494, 1746, 1074, 118, 1, 0, 24, 1537, 10126, 13110, 4050, 245, 1, 0, 43, 4642, 52129, 122521, 79396, 14111, 500, 1, 0, 77, 13745, 248494, 967644, 1126049, 425471, 46833, 1011, 1
Offset: 1
[1] [1]
[2] [0, 1]
[3] [0, 2, 1]
[4] [0, 3, 9, 1]
[5] [0, 5, 41, 24, 1]
[6] [0, 8, 150, 247, 55, 1]
[7] [0, 14, 494, 1746, 1074, 118, 1]
[8] [0, 24, 1537, 10126, 13110, 4050, 245, 1]
[9] [0, 43, 4642, 52129, 122521, 79396, 14111, 500, 1]
[10][0, 77, 13745, 248494, 967644, 1126049, 425471, 46833, 1011, 1]
.
The 5 irreducible permutations counted with T(5, 2) are 23451, 51234, 31524, 34512, and 45123.
A356264
Partition triangle read by rows, counting reducible permutations, refining triangle A356265.
Original entry on oeis.org
0, 0, 1, 0, 1, 2, 0, 1, 5, 3, 2, 0, 1, 9, 12, 15, 10, 2, 0, 1, 14, 23, 12, 47, 94, 11, 31, 24, 2, 0, 1, 20, 38, 48, 113, 293, 154, 137, 183, 409, 78, 63, 54, 2, 0, 1, 27, 60, 87, 49, 227, 738, 883, 451, 457, 670, 2157, 1007, 1580, 79, 605, 1520, 384, 127, 116, 2, 0
Offset: 0
[0] 0;
[1] 0;
[2] 1, 0;
[3] 1, 2, 0;
[4] 1, [5, 3], 2, 0;
[5] 1, [9, 12], [15, 10], 2, 0;
[6] 1, [14, 23, 12], [ 47, 94, 11], [31, 24], 2, 0;
[7] 1, [20, 38, 48], [113, 293, 154, 137], [183, 409, 78], [63, 54], 2, 0;
Summing the bracketed terms reduces the triangle to A356265.
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import collections
def reducible(p) -> bool: # p is a Sage-Permutation
return any(i for i in range(1, p.size())
if all(p(j) < p(k)
for j in range(1, i + 1)
for k in range(i + 1, p.size() + 1) ) )
def void(L) -> bool: return True
def perm_red_stats(n: int, part_costraint, lehmer_constraint):
res = collections.defaultdict(int)
for p in Permutations(n):
if not part_costraint(p): continue
l: list[int] = p.to_lehmer_code()
if lehmer_constraint(l):
c: list[int] = [l.count(i) for i in range(len(p)) if i in l]
res[Partition(reversed(sorted(c)))] += 1
return sorted(res.items(), key=lambda x: len(x[0]))
@cache
def A356264_row(n: int) -> list[int]:
if n < 2: return [0]
return [v[1] for v in perm_red_stats(n, reducible, void)] + [0]
def A356264(n: int, k: int) -> int:
return A356264_row(n)[k]
for n in range(0, 8): print(A356264_row(n))
A356114
Number of irreducible permutations of n with partition type [2, 1, 1, ..., 1] (with '1' taken n - 2 times).
Original entry on oeis.org
0, 0, 0, 2, 9, 24, 55, 118, 245, 500, 1011, 2034, 4081, 8176, 16367, 32750, 65517, 131052, 262123, 524266, 1048553, 2097128, 4194279, 8388582, 16777189, 33554404, 67108835, 134217698, 268435425, 536870880, 1073741791, 2147483614, 4294967261, 8589934556, 17179869147
Offset: 0
a(4) = 9 = card({2413, 2431, 3142, 3241, 3421, 4132, 4213, 4231, 4312}). The other two permutations of type [2, 1, 1], 1432 and 3214, are reducible. That there are 11 permutations of type [2, 1, 1] we know from Euler's triangle A173018 or from its refined form A355777.
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seq(`if`(n < 3, 0, combinat:-eulerian1(n, n - 2) - 2), n = 0..34);
Showing 1-3 of 3 results.
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