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A345162 Number of integer partitions of n with no alternating permutation covering an initial interval of positive integers.

%I A345162 #20 Jan 31 2024 20:50:30
%S A345162 0,0,1,1,1,2,2,3,3,5,6,6,8,10,11,15,16,18,23,27,30,35,41,47,54,62,71,
%T A345162 82,92,103,121,137,151,173,195,220,248,277,311,350,393,435,488,546,
%U A345162 605,678,754,835,928,1029,1141,1267,1400,1544,1712,1891,2081,2298,2533,2785,3068
%N A345162 Number of integer partitions of n with no alternating permutation covering an initial interval of positive integers.
%C A345162 A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no alternating permutations, even though it has anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).
%C A345162 Sequences covering an initial interval (patterns) are counted by A000670 and ranked by A333217.
%H A345162 Andrew Howroyd, <a href="/A345162/b345162.txt">Table of n, a(n) for n = 0..500</a>
%F A345162 a(n) = A000009(n) - A345163(n). - _Andrew Howroyd_, Jan 31 2024
%e A345162 The a(2) = 1 through a(10) = 6 partitions:
%e A345162   11  111  1111  2111   21111   2221     221111    22221      32221
%e A345162                  11111  111111  211111   2111111   321111     222211
%e A345162                                 1111111  11111111  2211111    3211111
%e A345162                                                    21111111   22111111
%e A345162                                                    111111111  211111111
%e A345162                                                               1111111111
%t A345162 normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t A345162 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A345162 Table[Length[Select[IntegerPartitions[n],normQ[#]&&Select[Permutations[#],wigQ[#]&]=={}&]],{n,0,15}]
%o A345162 (PARI) P(n,m)={Vec(1/prod(k=1, m, 1-y*x^k, 1+O(x*x^n)))}
%o A345162 a(n) = {(n >= 2) + sum(k=2, (sqrtint(8*n+1)-1)\2, my(r=n-binomial(k+1,2), v=P(r, k)); sum(i=1, min(k,2*r\k), sum(j=k-1, (2*r-(k-1)*(i-1))\(i+1), my(p=(j+k+(i==1||i==k))\2); if(p*i<=r, polcoef(v[r-p*i+1],j-p)) )))} \\ _Andrew Howroyd_, Jan 31 2024
%Y A345162 The complement in covering partitions is counted by A345163.
%Y A345162 Not requiring normality gives A345165, ranked by A345171.
%Y A345162 The separable case is A345166.
%Y A345162 A000041 counts integer partitions.
%Y A345162 A000670 counts patterns, ranked by A333217.
%Y A345162 A001250 counts alternating permutations.
%Y A345162 A003242 counts anti-run compositions.
%Y A345162 A005649 counts anti-run patterns.
%Y A345162 A025047 counts alternating or wiggly compositions, directed A025048/A025049.
%Y A345162 A325534 counts separable partitions, ranked by A335433.
%Y A345162 A325535 counts inseparable partitions, ranked by A335448.
%Y A345162 A344604 counts alternating compositions with twins.
%Y A345162 A344605 counts alternating patterns with twins.
%Y A345162 A345164 counts alternating permutations of prime indices.
%Y A345162 A345170 counts partitions with a alternating permutation, ranked by A345172.
%Y A345162 Cf. A000070, A103919, A335126, A344614, A344615, A344653, A344654, A344740, A344742, A345167, A345168, A345192, A348609.
%K A345162 nonn
%O A345162 0,6
%A A345162 _Gus Wiseman_, Jun 12 2021
%E A345162 a(26) onwards from _Andrew Howroyd_, Jan 31 2024