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' *)
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 *)
Showing 1-2 of 2 results.
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