A355777 Partition triangle read by rows. A statistic of permutations given by their Lehmer code refining Euler's triangle A173018.

1, 1, 1, 1, 1, 4, 1, 1, 7, 4, 11, 1, 1, 11, 15, 32, 34, 26, 1, 1, 16, 26, 15, 76, 192, 34, 122, 180, 57, 1, 1, 22, 42, 56, 156, 474, 267, 294, 426, 1494, 496, 423, 768, 120, 1, 1, 29, 64, 98, 56, 288, 1038, 1344, 768, 855, 1206, 5142, 2829, 5946, 496, 2127, 9204, 4288, 1389, 2904, 247, 1

Offset

0,6

Comments

An exposition of the theory is in Hivert et al. (see the table p. 4), test data can be found in the Statistics Database at St000275.

The ordering of the partitions is defined in A080577. See the comments in A356116 for the definition of the terms 'partition triangle' and 'reduced partition triangle'.

An alternative representation is Tom Copeland's A145271 which has a faster Maple program. The Sage program below, on the other hand, explicitly describes the combinatorial construction and shows how the permutations are bundled into partitions via the Lehmer code.

Links

Table of n, a(n) for n=0..66.

Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker and Amanda Welch, Homomesies on permutations -- an analysis of maps and statistics in the FindStat database, arXiv:2206.13409 [math.CO], 2022. (Def. 4.20 and Prop. 4.22.)

Florent Hivert, Jean-Christophe Novelli and Jean-Yves Thibon, Multivariate generalizations of the Foata-Schützenberger equidistribution, arXiv:math/0605060 [math.CO], 2006.

Florent Hivert, Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition, Statistics Database St000275, 2015.

Peter Luschny, Permutations with Lehmer, a SageMath Jupyter Notebook.

Wikipedia, Lehmer code.

Example

The table T(n, k) begins:
[0] 1;
[1] 1;
[2] 1, 1;
[3] 1, 4,            1;
[4] 1, [7, 4],       11,                   1;
[5] 1, [11, 15],     [32, 34],             26,               1;
[6] 1, [16, 26, 15], [76, 192, 34],        [122, 180],       57,         1;
[7] 1, [22, 42, 56], [156, 474, 267, 294], [426, 1494, 496], [423, 768], 120, 1;
Summing the bracketed terms reduces the triangle to Euler's triangle A173018.
.
The Lehmer mapping of the permutations to the partitions, case n = 4, k = 1:
   1243, 1324, 1423, 2134, 2341, 3124, 4123 map to the partition [3, 1] and
   1342, 2143, 2314, 3412 map to the partition [2, 2]. Thus A173018(4, 1) = 7 + 4 = 11.
.
The cardinality of the preimage of the partitions, i.e. the number of permutations which map to the same partition, are the terms of the sequence. Here row 6:
[6] => 1
[5, 1] => 16
[4, 2] => 26
[3, 3] => 15
[4, 1, 1] => 76
[3, 2, 1] => 192
[2, 2, 2] => 34
[3, 1, 1, 1] => 122
[2, 2, 1, 1] => 180
[2, 1, 1, 1, 1] => 57
[1, 1, 1, 1, 1, 1] => 1

Prog

(SageMath)
import collections
@cached_function
def eulerian_stat(n):
    res = collections.defaultdict(int)
    for p in Permutations(n):
        c = p.to_lehmer_code()
        l = [c.count(i) for i in range(len(p)) if i in c]
        res[Partition(reversed(sorted(l)))] += 1
    return sorted(res.items(), key=lambda x: len(x[0]))
@cached_function
def A355777_row(n): return [v[1] for v in eulerian_stat(n)]
def A355777(n, k): return A355777_row(n)[k]
for n in range(8): print(A355777_row(n))

Crossrefs

Cf. A000295 (subdiagonal), A000124 (column 2), A000142 (row sums), A000041 (row lengths).

Cf. A179454 (permutation trees), A123125 and A173018 (Eulerian numbers), A145271 (variant).

Sequence in context: A010321 A046550 A363975 * A223489 A016521 A146880

Adjacent sequences: A355774 A355775 A355776 * A355778 A355779 A355780

Keyword

nonn,tabf

Author

Peter Luschny, Jul 16 2022

Status

approved