cp's OEIS Frontend

A360674 Number of integer partitions of 2n whose left half (exclusive) and right half (inclusive) both sum to n.

%I A360674 #13 Mar 09 2023 20:03:32
%S A360674 1,1,3,4,7,6,12,9,16,15,21,16,34,22,33,36,47,36,62,44,75,68,78,68,120,
%T A360674 93,113,117,151,122,195,148,209,197,220,226,315,249,304,309,402,332,
%U A360674 463,387,496,515,539,514,712,609,738,723,845,774,983,914,1111
%N A360674 Number of integer partitions of 2n whose left half (exclusive) and right half (inclusive) both sum to n.
%C A360674 Of course, only one of the two conditions is necessary.
%F A360674 a(n) = A360672(2n,n).
%e A360674 The a(1) = 1 through a(6) = 12 partitions:
%e A360674   (11)  (22)    (33)      (44)        (55)          (66)
%e A360674         (211)   (321)     (422)       (532)         (633)
%e A360674         (1111)  (21111)   (431)       (541)         (642)
%e A360674                 (111111)  (2222)      (32221)       (651)
%e A360674                           (22211)     (211111111)   (3333)
%e A360674                           (2111111)   (1111111111)  (33222)
%e A360674                           (11111111)                (33321)
%e A360674                                                     (42222)
%e A360674                                                     (222222)
%e A360674                                                     (2222211)
%e A360674                                                     (21111111111)
%e A360674                                                     (111111111111)
%e A360674 For example, the partition y = (3,2,2,2,1) has halves (3,2) and (2,2,1), both with sum 5, so y is counted under a(5).
%t A360674 Table[Length[Select[IntegerPartitions[2n], Total[Take[#,Floor[Length[#]/2]]]==n&]],{n,0,15}]
%o A360674 (Python)
%o A360674 def accel_asc(n):
%o A360674     a = [0 for i in range(n + 1)]
%o A360674     k = 1
%o A360674     y = n - 1
%o A360674     while k != 0:
%o A360674         x = a[k - 1] + 1
%o A360674         k -= 1
%o A360674         while 2 * x <= y:
%o A360674             a[k] = x
%o A360674             y -= x
%o A360674             k += 1
%o A360674         l = k + 1
%o A360674         while x <= y:
%o A360674             a[k] = x
%o A360674             a[l] = y
%o A360674             yield a[:k + 2]
%o A360674             x += 1
%o A360674             y -= 1
%o A360674         a[k] = x + y
%o A360674         y = x + y - 1
%o A360674         yield a[:k + 1]
%o A360674 for y in range(1000):
%o A360674     num = 0
%o A360674     for x in accel_asc(2*y):
%o A360674         stop = len(x)//2+1
%o A360674         if len(x) % 2 == 0:
%o A360674             stop -= 1
%o A360674         right = x[0:stop]
%o A360674         left = x[stop:]
%o A360674         if sum(right) == sum(left):
%o A360674             num += 1
%o A360674     print(y,num)
%o A360674 # _David Consiglio, Jr._, Mar 09 2023
%Y A360674 The even-length case is A000005.
%Y A360674 Central diagonal of A360672.
%Y A360674 These partitions have ranks A360953.
%Y A360674 A008284 counts partitions by length, row sums A000041.
%Y A360674 A359893 and A359901 count partitions by median.
%Y A360674 First for prime indices, second for partitions, third for prime factors:
%Y A360674 - A360676 gives left sum (exclusive), counted by A360672, product A361200.
%Y A360674 - A360677 gives right sum (exclusive), counted by A360675, product A361201.
%Y A360674 - A360678 gives left sum (inclusive), counted by A360675, product A347043.
%Y A360674 - A360679 gives right sum (inclusive), counted by A360672, product A347044.
%Y A360674 Cf. A237363, A360254, A360671, A360673, A360682.
%K A360674 nonn
%O A360674 0,3
%A A360674 _Gus Wiseman_, Mar 04 2023
%E A360674 More terms from _David Consiglio, Jr._, Mar 09 2023