cp's OEIS Frontend

A325685 Number of compositions of n whose distinct consecutive subsequences have different sums, and such that these sums cover an initial interval of positive integers.

%I A325685 #19 Feb 21 2022 14:55:30
%S A325685 1,1,1,3,1,5,3,5,3,9,1,9,5,7,5,11,1,13,5,9,5,13,3,13,7,9,5,17,1,17,5,
%T A325685 9,9,15,5,15,5,13,5,21,1,17,9,9,9,17,3,21,7,13,5,17,5,21,9,13,5,21,1,
%U A325685 21,9,11,13,19,5,17,5,17,5,29,1,21,9,9,13,17,5,25,7,17,7
%N A325685 Number of compositions of n whose distinct consecutive subsequences have different sums, and such that these sums cover an initial interval of positive integers.
%C A325685 A composition of n is a finite sequence of positive integers summing to n.
%C A325685 Compare to the definition of perfect partitions (A002033).
%H A325685 Fausto A. C. Cariboni, <a href="/A325685/b325685.txt">Table of n, a(n) for n = 0..100</a>
%H A325685 Fausto A. C. Cariboni, <a href="/A325685/a325685.txt">All compositions that yield a(n) for n = 1..100</a>, Feb 21 2022.
%e A325685 The distinct consecutive subsequences of (3,4,1,1) together with their sums are:
%e A325685    1: {1}
%e A325685    2: {1,1}
%e A325685    3: {3}
%e A325685    4: {4}
%e A325685    5: {4,1}
%e A325685    6: {4,1,1}
%e A325685    7: {3,4}
%e A325685    8: {3,4,1}
%e A325685    9: {3,4,1,1}
%e A325685 Because the sums are all different and cover {1...9}, it follows that (3,4,1,1) is counted under a(9).
%e A325685 The a(1) = 1 through a(9) = 9 compositions:
%e A325685   1   11   12    1111   113     132      1114      1133       1143
%e A325685            21           122     231      1222      3311       1332
%e A325685            111          221     111111   2221      11111111   2331
%e A325685                         311              4111                 3411
%e A325685                         11111            1111111              11115
%e A325685                                                               12222
%e A325685                                                               22221
%e A325685                                                               51111
%e A325685                                                               111111111
%t A325685 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Sort[Total/@Union[ReplaceList[#,{___,s__,___}:>{s}]]]==Range[n]&]],{n,0,15}]
%Y A325685 Cf. A000079, A002033, A103295, A126796, A143823, A169942, A325676, A325677, A325683, A325684.
%K A325685 nonn
%O A325685 0,4
%A A325685 _Gus Wiseman_, May 13 2019
%E A325685 a(21)-a(25) from _Jinyuan Wang_, Jun 26 2020
%E A325685 a(21)-a(25) corrected, a(26)-a(80) from _Fausto A. C. Cariboni_, Feb 21 2022