This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357611 #78 Dec 28 2024 10:18:36
%S A357611 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,
%T A357611 5,14,19,10,14,35,5,40,26,19,61,10,40,26,35,35,26,40,10,61,19,26,40,5,
%U A357611 35,14,10,19,14,5,1
%N A357611 A refinement of the Mahonian numbers (canonical ordering).
%C A357611 Let T(N,d) be a Mahonian number.
%C A357611 (1) Find all partitions k(1) >= k(2) >= ... >= k(N) = 0 of the number d with at most N-1 parts, such that k(i) - k(i+1) <= 1 and k(N) = 0.
%C A357611 (2) For each such partition, draw a ribbon Young diagram with N boxes at matrix (row-column) coordinates (k(i), k(i) + i - 1), i = 1, ..., N.
%C A357611 (3) For each ribbon diagram, count all standard Young skew tableaux.
%C A357611 The numbers under (3) will add up to T(N,d), as proven in the cited reference. These refinements appear consecutively as subsequences in the sequence, ordered by increasing N and decreasing d from N(N-1)/2 to 0 for each N. The ordering within each subsequence is reverse-lexicographic by the partitions (1), i.e., from the largest k(1) down.
%C A357611 This ordering is called canonical because it corresponds to the ordering of generating polynomials (from the highest power down). Because of certain symmetries in the sequence (cf. the example), it is the same as increasing d from 0 to N(N-1)/2 and ordering the partitions lexicographically. For the converse convention, cf. A356802.
%C A357611 The sums of rows (3) are A008302. Because the latter is itself a triangle, arranged in the (N, d) plane, the present sequence is actually three-dimensional: each number in the triangle A008302 can be replaced by the row numbers (3), each of which is thus coordinatized by (N, d, m), the last coordinate being its position in the row.
%C A357611 The range of the m coordinate is the number of partitions satisfying the constraint (1). It is proven in the cited reference to be equal to the coefficient of q^d in the q-binomial theorem: coeff[q^d] Product_{k=1..N-1} (1 + q^k).
%C A357611 This is a different ordering of A060351 and A335845 because every standard Young ribbon diagram of a given shape corresponds to a permutation with a given descent set, see the example. - _Andrey Zabolotskiy_, Oct 08 2024
%C A357611 However, the orderings in A060351 and A335845 cannot be considered a refinement of Mahonians because the terms adding to a single Mahonian do not appear consecutively in them, beginning with N=5 in the example. - _Denis K. Sunko_, Dec 27 2024
%H A357611 Denis K. Sunko, <a href="https://arxiv.org/abs/2209.02523">Evaluation and spanning sets of confluent Vandermonde forms</a>, arXiv:2209.02523 [math-ph], 2022.
%H A357611 Denis K. Sunko, <a href="https://doi.org/10.1063/5.0075576">Evaluation and spanning sets of confluent Vandermonde forms</a>, J. Math. Phys. 63, 082101 (2022).
%e A357611 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).
%e A357611 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:
%e A357611 (A) ## (B) #
%e A357611 # ###
%e A357611 #
%e A357611 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).
%e A357611 Diagram (A) can be filled in as a standard Young diagram in 3 ways:
%e A357611 12 14 13
%e A357611 3 2 2
%e A357611 4 3 4
%e A357611 Diagram (B) can be filled in in 3 ways, too:
%e A357611 2 3 1
%e A357611 134 124 234
%e A357611 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!).
%e A357611 The data in triangular form are:
%e A357611 N, d
%e A357611 1, 0 1,
%e A357611 2, 1 1,
%e A357611 0 1,
%e A357611 3, 3 1,
%e A357611 2 2,
%e A357611 1 2,
%e A357611 0 1,
%e A357611 4, 6 1,
%e A357611 5 3,
%e A357611 4 5,
%e A357611 3 3, 3,
%e A357611 2 5,
%e A357611 1 3,
%e A357611 0 1,
%e A357611 5,10 1,
%e A357611 9 4,
%e A357611 8 9,
%e A357611 7 9, 6,
%e A357611 6 4, 16,
%e A357611 5 11, 11,
%e A357611 4 16, 4,
%e A357611 3 6, 9,
%e A357611 2 9,
%e A357611 1 4,
%e A357611 0 1,
%e A357611 6,15 1,
%e A357611 14 5,
%e A357611 13 14,
%e A357611 12 19, 10,
%e A357611 11 14, 35,
%e A357611 10 5, 40, 26,
%e A357611 9 19, 61, 10,
%e A357611 8 40, 26, 35,
%e A357611 7 35, 26, 40,
%e A357611 6 10, 61, 19,
%e A357611 5 26, 40, 5,
%e A357611 4 35, 14,
%e A357611 3 10, 19,
%e A357611 2 14,
%e A357611 1 5,
%e A357611 0 1
%e A357611 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.
%o A357611 (SageMath)
%o A357611 def possible_classes(n, degree):
%o A357611 for stub in Partitions(degree, max_part = n-1, max_length = n-1, min_slope = -1):
%o A357611 cls = list(stub)
%o A357611 if cls:
%o A357611 if cls[-1] < 2:
%o A357611 yield cls + (n - len(cls)) * [0]
%o A357611 else:
%o A357611 yield n * [0]
%o A357611 #
%o A357611 def count_tableaux(cls):
%o A357611 inner = []
%o A357611 outer = [1]
%o A357611 right = 1
%o A357611 previous_part = cls[0]
%o A357611 for part in cls[1:]:
%o A357611 if part == previous_part:
%o A357611 right += 1
%o A357611 outer[-1] = right
%o A357611 else:
%o A357611 previous_part = part
%o A357611 outer += [right]
%o A357611 if right > 1:
%o A357611 inner += [right-1]
%o A357611 outer.reverse()
%o A357611 inner.reverse()
%o A357611 return StandardSkewTableaux([outer, inner]).cardinality()
%o A357611 #
%o A357611 def refine_mahonian(N, d, total = False):
%o A357611 """
%o A357611 Eq. (50) in DOI:10.48550/arXiv.2209.02523 was
%o A357611 generated by the call refine_mahonian(8, 16, True).
%o A357611 """
%o A357611 res = []
%o A357611 for cls in possible_classes(N,d):
%o A357611 res += [count_tableaux(cls)]
%o A357611 if total:
%o A357611 res = (res, sum(res)) # the sum should be T(N, d)
%o A357611 return res
%o A357611 #
%o A357611 def refine_mahonians_table(Nmax, total = False, canonical = True):
%o A357611 res = []
%o A357611 for N in range(1, Nmax + 1):
%o A357611 r = []
%o A357611 if canonical:
%o A357611 ordering = range(N * (N - 1) // 2, -1, -1)
%o A357611 else:
%o A357611 ordering = range(N * (N - 1) // 2 + 1)
%o A357611 for d in ordering:
%o A357611 r += [refine_mahonian(N, d, total = total)]
%o A357611 res += [r]
%o A357611 return res
%o A357611 #
%o A357611 def refine_mahonians(Nmax, canonical = True):
%o A357611 """
%o A357611 Nmax = 6, canonical = True gives seq. A357611 in the OEIS.
%o A357611 Nmax = 6, canonical = False gives seq. A356802 in the OEIS.
%o A357611 """
%o A357611 return flatten(refine_mahonians_table(Nmax, total = False, canonical = canonical))
%o A357611 (SageMath)
%o A357611 from collections import Counter
%o A357611 def part(n, descents):
%o A357611 r = tuple(sum(i <= d for d in descents) for i in (1..n))
%o A357611 return (sum(r), r) # replace sum(r) by -sum(r) to obtain A356802 instead
%o A357611 def row(n):
%o A357611 return [x[1] for x in sorted(Counter((part(n, p.descents()) for p in Permutations(n))).items())]
%o A357611 print(sum([row(n) for n in (1..6)], [])) # _Andrey Zabolotskiy_, Oct 19 2024
%Y A357611 Cf. A008302, A356802, A060351, A335845, A360308.
%K A357611 nonn,tabf
%O A357611 1,5
%A A357611 _Denis K. Sunko_, Oct 06 2022