A356262
Partition triangle read by rows counting the irreducible permutations sorted by the partition type of their Lehmer code.
Original entry on oeis.org
1, 1, 0, 1, 0, 2, 1, 0, 2, 1, 9, 1, 0, 2, 3, 24, 17, 24, 1, 0, 2, 3, 3, 98, 29, 23, 156, 91, 55, 1, 0, 2, 8, 4, 181, 43, 157, 113, 1085, 243, 418, 714, 360, 118, 1, 0, 2, 7, 11, 4, 300, 61, 317, 461, 398, 2985, 536, 1822, 4366, 417, 7684, 1522, 3904, 2788, 1262, 245, 1
Offset: 0
[0] 1;
[1] 1;
[2] 0, 1;
[3] 0, 2, 1;
[4] 0, [2, 1], 9, 1;
[5] 0, [2, 3], [24, 17], 24, 1;
[6] 0, [2, 3, 3], [98, 29, 23], [156, 91], 55, 1;
[7] 0, [2, 8, 4], [181, 43, 157, 113], [1085, 243, 418], [714, 360], 118, 1;
Summing the bracketed terms reduces the triangle to A356263 .
.
The Lehmer mapping of the irreducible permutations to the partitions, case n = 4, k = 1: 2341 and 4123 map to the partition [3, 1], and 3412 map to the partition [2, 2]. Thus A356263(4, 1) = 2 + 1 = 3. Compare with the example in A355777.
.
The partition mapping of row 4:
[4] => 0
[3, 1] => 2
[2, 2] => 1
[2, 1, 1] => 9
[1, 1, 1, 1] => 1
-
import collections
def reducible(p) -> bool:
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 perm_irreducible_stats(n: int):
res = collections.defaultdict(int)
for p in Permutations(n):
if reducible(p): continue
l = p.to_lehmer_code()
c = [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]))
@cached_function
def A356262_row(n):
if n <= 1: return [1]
return [0] + [v[1] for v in perm_irreducible_stats(n)]
def A356262(n, k): return A356262_row(n)[k]
for n in range(0, 8): print(A356262_row(n))
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.
-
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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