A357611 A refinement of the Mahonian numbers (canonical ordering).
1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 3, 5, 3, 1, 1, 4, 9, 9, 6, 4, 16, 11, 11, 16, 4, 6, 9, 9, 4, 1, 1, 5, 14, 19, 10, 14, 35, 5, 40, 26, 19, 61, 10, 40, 26, 35, 35, 26, 40, 10, 61, 19, 26, 40, 5, 35, 14, 10, 19, 14, 5, 1
Offset: 1
Examples
The first nontrivial terms in the sequence are a(11) = a(12) = 3, corresponding to the refinement T(4, 3) = 6 = 3 + 3. The terms from a(1) to a(10) are the Mahonian numbers themselves, because the refinement is trivial for them (there is only one partition satisfying the given constraints).
Specifically, the row T(N, d) with N=4 and d=3 corresponds to Young ribbon diagrams with 4 cells such that the sum of the row indices of cells (with the top row having index 0) is equal to 3. There are two such diagrams:
(A) ## (B) #
# ###
#
3 = 2+1+0+0 = 1+1+1+0 are the corresponding integer partitions, which are referenced in the Comments section, listed in the lexicographic order. These partitions have descent sets (indices of elements followed by a smaller element) {1,2} and {3}, respectively (they both sum up to 3, necessarily).
Diagram (A) can be filled in as a standard Young diagram in 3 ways:
12 14 13
3 2 2
4 3 4
Diagram (B) can be filled in in 3 ways, too:
2 3 1
134 124 234
Thus, the row T(4, 3) is 3, 3. These standard Young ribbon diagrams, when read bottom-left to top-right, become permutations of 1234 with major index 3, namely 4312, 3214, 4213 with the descent set {1,2} and 1342, 1243, 2341 with the descent set {3} (same descent sets as those of the corresponding partitions!).
The data in triangular form are:
N, d
1, 0 1,
2, 1 1,
0 1,
3, 3 1,
2 2,
1 2,
0 1,
4, 6 1,
5 3,
4 5,
3 3, 3,
2 5,
1 3,
0 1,
5,10 1,
9 4,
8 9,
7 9, 6,
6 4, 16,
5 11, 11,
4 16, 4,
3 6, 9,
2 9,
1 4,
0 1,
6,15 1,
14 5,
13 14,
12 19, 10,
11 14, 35,
10 5, 40, 26,
9 19, 61, 10,
8 40, 26, 35,
7 35, 26, 40,
6 10, 61, 19,
5 26, 40, 5,
4 35, 14,
3 10, 19,
2 14,
1 5,
0 1
One can check the generating function for the number of terms in a row, e.g., for N = 4: (1 + q)(1 + q^2)(1 + q^3) = q^6 + q^5 + q^4 + 2q^3 + q^2 + q + 1.
Links
- Denis K. Sunko, Evaluation and spanning sets of confluent Vandermonde forms, arXiv:2209.02523 [math-ph], 2022.
- Denis K. Sunko, Evaluation and spanning sets of confluent Vandermonde forms, J. Math. Phys. 63, 082101 (2022).
Programs
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SageMath
def possible_classes(n, degree): for stub in Partitions(degree, max_part = n-1, max_length = n-1, min_slope = -1): cls = list(stub) if cls: if cls[-1] < 2: yield cls + (n - len(cls)) * [0] else: yield n * [0] # def count_tableaux(cls): inner = [] outer = [1] right = 1 previous_part = cls[0] for part in cls[1:]: if part == previous_part: right += 1 outer[-1] = right else: previous_part = part outer += [right] if right > 1: inner += [right-1] outer.reverse() inner.reverse() return StandardSkewTableaux([outer, inner]).cardinality() # def refine_mahonian(N, d, total = False): """ Eq. (50) in DOI:10.48550/arXiv.2209.02523 was generated by the call refine_mahonian(8, 16, True). """ res = [] for cls in possible_classes(N,d): res += [count_tableaux(cls)] if total: res = (res, sum(res)) # the sum should be T(N, d) return res # def refine_mahonians_table(Nmax, total = False, canonical = True): res = [] for N in range(1, Nmax + 1): r = [] if canonical: ordering = range(N * (N - 1) // 2, -1, -1) else: ordering = range(N * (N - 1) // 2 + 1) for d in ordering: r += [refine_mahonian(N, d, total = total)] res += [r] return res # def refine_mahonians(Nmax, canonical = True): """ Nmax = 6, canonical = True gives seq. A357611 in the OEIS. Nmax = 6, canonical = False gives seq. A356802 in the OEIS. """ return flatten(refine_mahonians_table(Nmax, total = False, canonical = canonical)) -
SageMath
from collections import Counter def part(n, descents): r = tuple(sum(i <= d for d in descents) for i in (1..n)) return (sum(r), r) # replace sum(r) by -sum(r) to obtain A356802 instead def row(n): return [x[1] for x in sorted(Counter((part(n, p.descents()) for p in Permutations(n))).items())] print(sum([row(n) for n in (1..6)], [])) # Andrey Zabolotskiy, Oct 19 2024
Comments