cp's OEIS Frontend

A351983 Number of integer compositions of n with exactly one part above the diagonal.

%I A351983 #13 Jan 02 2023 21:55:17
%S A351983 0,0,1,2,5,9,18,35,67,131,257,505,996,1973,3915,7781,15486,30855,
%T A351983 61527,122764,245069,489412,977673,1953515,3904108,7803545,15599618,
%U A351983 31187269,62355347,124679883,249310255,498540890,996953659,1993701032,3987069747,7973603891
%N A351983 Number of integer compositions of n with exactly one part above the diagonal.
%H A351983 Andrew Howroyd, <a href="/A351983/b351983.txt">Table of n, a(n) for n = 0..1000</a>
%e A351983 The a(2) = 1 through a(6) = 18 compositions:
%e A351983   (2)  (3)   (4)    (5)     (6)
%e A351983        (21)  (13)   (14)    (15)
%e A351983              (22)   (32)    (42)
%e A351983              (31)   (41)    (51)
%e A351983              (211)  (131)   (114)
%e A351983                     (212)   (132)
%e A351983                     (221)   (141)
%e A351983                     (311)   (213)
%e A351983                     (2111)  (222)
%e A351983                             (312)
%e A351983                             (321)
%e A351983                             (411)
%e A351983                             (1311)
%e A351983                             (2112)
%e A351983                             (2121)
%e A351983                             (2211)
%e A351983                             (3111)
%e A351983                             (21111)
%t A351983 pless[y_]:=Length[Select[Range[Length[y]],#<y[[#]]&]];
%t A351983 Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],pless[#]==1&]],{n,0,10}]
%o A351983 (PARI)
%o A351983 S(v,u,c=0)={vector(#v, k, c + sum(i=1, k-1, v[k-i]*u[i]))}
%o A351983 seq(n)={my(v=vector(1+n), s=0); v[1]=1; for(i=1, n, v=S(v, vector(n, j, if(j>i,'x,1)), O(x^2)); s+=apply(p->polcoef(p,1), v)); s} \\ _Andrew Howroyd_, Jan 02 2023
%Y A351983 The version for permutations is A000295, weak A057427.
%Y A351983 The version for partitions is A002620, weak A001477.
%Y A351983 The weak version is A177510.
%Y A351983 The version for fixed points is A240736, nonfixed A352520.
%Y A351983 This is column k = 1 of A352524; column k = 0 is A008930.
%Y A351983 A238349 counts compositions by fixed points, first column A238351.
%Y A351983 A352521 counts compositions by strong nonexcedances, first column A219282.
%Y A351983 A352522 counts compositions by weak nonexcedances, first column A238874.
%Y A351983 A352523 counts compositions by nonfixed points, first column A010054.
%Y A351983 A352524 counts compositions by strong excedances, first column A008930.
%Y A351983 A352525 counts compositions by weak excedances, first column A177510.
%Y A351983 Cf. A088218, A098825, A115994, A238352, A330644, A352516.
%K A351983 nonn
%O A351983 0,4
%A A351983 _Gus Wiseman_, Apr 02 2022
%E A351983 Terms a(21) and beyond from _Andrew Howroyd_, Jan 02 2023