A248476
Number of pairs (not necessarily successors) of partitions of n that are incomparable in dominance (natural, majorization) ordering.
Original entry on oeis.org
0, 0, 0, 0, 0, 4, 8, 30, 70, 170, 340, 770, 1424, 2810, 5166, 9542, 16614, 29596, 49952, 85610, 141604, 234622, 379218, 616008, 976134, 1549134, 2418768, 3771252, 5795300, 8903306, 13497384, 20438432, 30630108, 45789134, 67857566, 100346480, 147170162, 215341690
Offset: 1
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Table[Count[ Flatten[Outer[dominant , Partitions[n], Partitions[n], 1]], 0], {n, 24}] (* see A248475 for definition of 'dominant' *)
A248475
Number of pairs of partitions of n that are successors in reverse lexicographic order, but incomparable in dominance (natural, majorization) ordering.
Original entry on oeis.org
0, 0, 0, 0, 0, 2, 3, 4, 6, 9, 12, 17, 22, 30, 39, 51, 65, 85, 107, 136, 171, 216, 268, 335, 413, 512, 629, 772, 941, 1151, 1396, 1694, 2046, 2471, 2969, 3569, 4271, 5110, 6093, 7258, 8620, 10235, 12113, 14325, 16902, 19925, 23434, 27540, 32296, 37842, 44260, 51715, 60322, 70306, 81805
Offset: 1
The successor pair (3,1,1,1) and (2,2,2) are incomparable in dominance ordering, and so are their transposes (4,1,1) and (3,3) and these are the two only pairs for n=6, hence a(6)=2.
- Ian G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1979, pp. 6-8.
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Needs["Combinatorica`"];
dominant[par1_?PartitionQ,par2_?PartitionQ]:= Block[{le=Max[Length[par1],Length[par2]],acc},
acc=Accumulate[PadRight[par1,le]]-Accumulate[PadRight[par2,le]];Which[Min[acc]===0&&Max[acc]>=0,1,Min[acc]<=0&&Max[acc]===0,-1,True,0]];
Table[Count[Apply[dominant, Partition[Partitions[n], 2,1], 1],0], {n,40}]
A265506
Number of pairs (p,q) of partitions of n into distinct parts such that p majorizes q in the dominance order.
Original entry on oeis.org
1, 1, 1, 3, 3, 6, 10, 15, 21, 35, 54, 75, 115, 161, 238, 349, 486, 673, 972, 1323, 1840, 2562, 3478, 4711, 6407, 8624, 11533, 15502, 20574, 27194, 36030, 47320, 61833, 81139, 105286, 136845, 177369, 228563, 293787, 377803, 483090, 616546, 785925, 997987
Offset: 0
a(3) = 3: (21,21), (3,21), (3,3).
a(4) = 3: (31,31), (4,31), (4,4).
a(5) = 6: (32,32), (41,32), (41,41), (5,32), (5,41), (5,5).
a(6) = 10: (321,321), (42,321), (42,42), (51,321), (51,42), (51,51), (6,321), (6,42), (6,51), (6,6).
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b:= proc(n, m, i, j, t) option remember; `if`(n0, b(n, m, i, j-1, true), 0)+
b(n, m, i-1, j, false)+b(n-i, m-j, max(0, min(n-i, i-1)),
max(0, min(m-j, j-1)), true))))
end:
a:= n-> b(n$4, true):
seq(a(n), n=0..40);
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b[n_, m_, i_, j_, t_] := b[n, m, i, j, t] = If[n < m, 0, If[n == 0, 1, If[i < 1, 0, If[t && j > 0, b[n, m, i, j-1, True], 0] + b[n, m, i-1, j, False] + b[n-i, m-j, Max[0, Min[n-i, i-1]], Max[0, Min[m-j, j-1]], True]]]]; a[n_] := b[n, n, n, n, True]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 05 2017, translated from Maple *)
Original entry on oeis.org
1, 4, 7, 12, 15, 20, 24, 31, 34, 39, 43, 50, 53, 59, 64, 75, 78, 83, 89
Offset: 1
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