cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A372119 Numbers k such that the k-th composition in standard order is not biquanimous.

This page as a plain text file.
%I A372119 #6 Apr 20 2024 10:51:33
%S A372119 1,2,4,5,6,7,8,9,12,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,
%T A372119 32,33,34,35,40,42,48,49,56,64,65,66,67,68,69,70,71,72,73,74,75,76,77,
%U A372119 78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96
%N A372119 Numbers k such that the k-th composition in standard order is not biquanimous.
%C A372119 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%C A372119 A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.
%e A372119 The terms and corresponding compositions begin:
%e A372119    1: (1)
%e A372119    2: (2)
%e A372119    4: (3)
%e A372119    5: (2,1)
%e A372119    6: (1,2)
%e A372119    7: (1,1,1)
%e A372119    8: (4)
%e A372119    9: (3,1)
%e A372119   12: (1,3)
%e A372119   16: (5)
%e A372119   17: (4,1)
%e A372119   18: (3,2)
%e A372119   19: (3,1,1)
%e A372119   20: (2,3)
%e A372119   21: (2,2,1)
%e A372119   22: (2,1,2)
%e A372119   23: (2,1,1,1)
%t A372119 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A372119 Select[Range[0,100],!MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&]
%Y A372119 The unordered complement is A357976, counted by A002219.
%Y A372119 The unordered version is A371731, counted by A371795, even case A006827.
%Y A372119 These compositions are counted by A371956.
%Y A372119 The complement is A372120, counted by A064914.
%Y A372119 A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
%Y A372119 A321142 and A371794 count non-biquanimous strict partitions.
%Y A372119 A371791 counts biquanimous sets, differences A232466.
%Y A372119 A371792 counts non-biquanimous sets, differences A371793.
%Y A372119 Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.
%K A372119 nonn
%O A372119 1,2
%A A372119 _Gus Wiseman_, Apr 20 2024