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

0, 0, 1, 2, 5, 9, 18, 35, 67, 131, 257, 505, 996, 1973, 3915, 7781, 15486, 30855, 61527, 122764, 245069, 489412, 977673, 1953515, 3904108, 7803545, 15599618, 31187269, 62355347, 124679883, 249310255, 498540890, 996953659, 1993701032, 3987069747, 7973603891

Offset

0,4

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Example

The a(2) = 1 through a(6) = 18 compositions:
  (2)  (3)   (4)    (5)     (6)
       (21)  (13)   (14)    (15)
             (22)   (32)    (42)
             (31)   (41)    (51)
             (211)  (131)   (114)
                    (212)   (132)
                    (221)   (141)
                    (311)   (213)
                    (2111)  (222)
                            (312)
                            (321)
                            (411)
                            (1311)
                            (2112)
                            (2121)
                            (2211)
                            (3111)
                            (21111)

Mathematica

pless[y_]:=Length[Select[Range[Length[y]], #<y[[#]]&]];
Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], pless[#]==1&]], {n, 0, 10}]

Prog

(PARI)
S(v, u, c=0)={vector(#v, k, c + sum(i=1, k-1, v[k-i]*u[i]))}
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

Crossrefs

The version for permutations is A000295, weak A057427.

The version for partitions is A002620, weak A001477.

The weak version is A177510.

The version for fixed points is A240736, nonfixed A352520.

This is column k = 1 of A352524; column k = 0 is A008930.

A238349 counts compositions by fixed points, first column A238351.

A352521 counts compositions by strong nonexcedances, first column A219282.

A352522 counts compositions by weak nonexcedances, first column A238874.

A352523 counts compositions by nonfixed points, first column A010054.

A352524 counts compositions by strong excedances, first column A008930.

A352525 counts compositions by weak excedances, first column A177510.

Cf. A088218, A098825, A115994, A238352, A330644, A352516.

Sequence in context: A117353 A103422 A217210 * A247322 A348473 A097281

Adjacent sequences: A351980 A351981 A351982 * A351984 A351985 A351986

Keyword

nonn

Author

Gus Wiseman, Apr 02 2022

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 02 2023

Status

approved