A238349 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k parts p at position p (fixed points), n>=0, 0<=k<=n.

1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 3, 4, 1, 0, 0, 6, 7, 3, 0, 0, 0, 11, 16, 4, 1, 0, 0, 0, 22, 29, 12, 1, 0, 0, 0, 0, 42, 60, 23, 3, 0, 0, 0, 0, 0, 82, 120, 47, 7, 0, 0, 0, 0, 0, 0, 161, 238, 100, 12, 1, 0, 0, 0, 0, 0, 0, 316, 479, 198, 30, 1, 0, 0, 0, 0, 0, 0, 0, 624, 956, 404, 61, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1235, 1910, 818, 126, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,7

Comments

T(n*(n+3)/2,n) = A227682(n).

From Vaclav Kotesovec, Sep 07 2014: (Start)

In general, column k is asymptotic to c(k) * 2^n. The constants c(k) numerically:

c(0) = 0.144394047543301210639449860964615390044455952420342... = A048651/2

c(1) = 0.231997216225445223894202367545783700531838988546098... = c(0)*A065442

c(2) = 0.104261929557371534733906196116707679501974368826074...

c(3) = 0.017956317806894073430249112172514186063327165575720...

c(4) = 0.001343254222922697613125145839110293324517874530073...

c(5) = 0.000046459767012163920051487037952792359225887287888...

c(6) = 0.000000768651747857094917953943327540619110335556499...

c(7) = 0.000000006200599904985793344094393321042983316604040...

c(8) = 0.000000000024656652167851516076173236693314090168122...

c(9) = 0.000000000000048633746319332356416193899916110113745...

c(10)= 0.000000000000000047750743608910618576944191079881479...

c(20)= 1.05217230403079700467566...*10^(-63)

For big k is c(k) ~ m * 2^(-k*(k+1)/2), where m = 1/(4*c(0)) = 1/(2*A048651) = 1.7313733097275318...

(End)

References

M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

Links

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10010 (rows 0..140, flattened)

M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

Example

Triangle starts:
00:  1,
01:  0, 1,
02:  1, 1, 0,
03:  2, 1, 1, 0,
04:  3, 4, 1, 0, 0,
05:  6, 7, 3, 0, 0, 0,
06:  11, 16, 4, 1, 0, 0, 0,
07:  22, 29, 12, 1, 0, 0, 0, 0,
08:  42, 60, 23, 3, 0, 0, 0, 0, 0,
09:  82, 120, 47, 7, 0, 0, 0, 0, 0, 0,
10:  161, 238, 100, 12, 1, 0, 0, 0, 0, 0, 0,
11:  316, 479, 198, 30, 1, 0, 0, 0, 0, 0, 0, 0,
12:  624, 956, 404, 61, 3, 0, 0, 0, 0, 0, 0, 0, 0,
13:  1235, 1910, 818, 126, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0,
14:  2449, 3817, 1652, 258, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
15:  4864, 7633, 3319, 537, 30, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
...
From Gus Wiseman, Apr 03 2022: (Start)
Row n = 5 counts the following compositions (empty columns indicated by dots):
  (5)     (14)     (113)   .  .  .
  (23)    (32)     (122)
  (41)    (131)    (1211)
  (212)   (221)
  (311)   (1112)
  (2111)  (1121)
          (11111)
(End)

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, expand(
       add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..15);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[b[n-j, i+1]*If[i == j, x, 1], {j, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)
pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], pq[#]==k&]], {n, 0, 9}, {k, 0, n}] (* Gus Wiseman, Apr 03 2022 *)

Crossrefs

Row sums are A011782.

Columns k=0-10 give: A238351, A240736, A240737, A240738, A240739, A240740, A240741, A240742, A240743, A240744, A240745.

The version for permutations is A008290.

The version with all zeros removed is A238350.

The version for reversed partitions is A238352.

The corresponding rank statistic is A352512, nonfixed A352513.

The version for nonfixed points is A352523, A352520 (k=1).

Below: comps = compositions, first = column k=0, stat = rank statistic.

- A352521 counts comps by strong nonexcedances, first A219282, stat A352514.

- A352522 counts comps by weak nonexcedances, first A238874, stat A352515.

- A352524 counts comps by strong excedances, first A008930, stat A352516.

- A352525 counts comps by weak excedances, A177510 (k=1), stat A352517.

Cf. A008292, A088218, A098825, A227682, A352487.

Sequence in context: A111571 A051509 A124816 * A318754 A318758 A334192

Adjacent sequences: A238346 A238347 A238348 * A238350 A238351 A238352

Keyword

nonn,tabl

Author

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Status

approved