cp's OEIS Frontend

A343941 Number of strict integer partitions of 2n with reverse-alternating sum 4.

%I A343941 #8 Jun 12 2021 06:09:02
%S A343941 0,0,1,0,1,2,3,3,4,5,7,8,10,11,14,15,18,20,23,25,29,31,35,38,42,45,50,
%T A343941 53,58,62,67,71,77,81,87,92,98,103,110,115,122,128,135,141,149,155,
%U A343941 163,170,178,185,194,201,210,218,227,235,245,253,263,272,282,291,302
%N A343941 Number of strict integer partitions of 2n with reverse-alternating sum 4.
%C A343941 The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.  This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts, so a(n) is the number of strict odd-length integer partitions of 2n whose conjugate has exactly 4 odd parts (first example). By conjugation, this is also the number partitions of 2n covering an initial interval and containing exactly four odd parts, one of which is the greatest (second example).
%e A343941 The a(2) = 1 through a(12) = 10 strict partitions (empty column indicated by dot, A..D = 10..13):
%e A343941   4   .  521   532   543   653   763     873     983     A93     BA3
%e A343941                631   642   752   862     972     A82     B92     CA2
%e A343941                      741   851   961     A71     B81     C91     DA1
%e A343941                                  64321   65421   65432   76432   76542
%e A343941                                          75321   75431   76531   86541
%e A343941                                                  76421   86431   87432
%e A343941                                                  86321   87421   87531
%e A343941                                                          97321   97431
%e A343941                                                                  98421
%e A343941                                                                  A8321
%e A343941 The a(2) = 1 through a(8) = 5 partitions covering an initial interval:
%e A343941   1111  .  32111   33211    33321     333221     543211      543321
%e A343941                    322111   332211    3322211    3332221     5432211
%e A343941                             3222111   32222111   33222211    33322221
%e A343941                                                  322222111   332222211
%e A343941                                                              3222222111
%t A343941 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A343941 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==4&]],{n,0,30,2}]
%Y A343941 The non-reverse non-strict version is A000710.
%Y A343941 The non-reverse version is A026810.
%Y A343941 The non-strict version is column k = 2 of A344610.
%Y A343941 This is column k = 2 of A344649.
%Y A343941 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A343941 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A343941 A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
%Y A343941 A124754 gives alternating sums of standard compositions (reverse: A344618).
%Y A343941 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A343941 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A343941 Cf. A000070, A000097, A003242, A006330, A027187, A119899, A152146, A239830, A325535, A344604, A344607, A344608, A344650, A344739.
%K A343941 nonn
%O A343941 0,6
%A A343941 _Gus Wiseman_, Jun 09 2021
%E A343941 More terms from _Bert Dobbelaere_, Jun 12 2021