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A298668 Number T(n,k) of set partitions of [n] into k blocks such that the absolute difference between least elements of consecutive blocks is always > 1; triangle T(n,k), n>=0, 0<=k<=ceiling(n/2), read by rows. 2
1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 3, 0, 1, 7, 2, 0, 1, 15, 12, 0, 1, 31, 50, 6, 0, 1, 63, 180, 60, 0, 1, 127, 602, 390, 24, 0, 1, 255, 1932, 2100, 360, 0, 1, 511, 6050, 10206, 3360, 120, 0, 1, 1023, 18660, 46620, 25200, 2520, 0, 1, 2047, 57002, 204630, 166824, 31920, 720 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

FORMULA

T(n,k) = (k-1)! * Stirling2(n-k+1,k) for k>0, T(n,0) = A000007(n).

T(n,k) = Sum_{j=0..k-1} (-1)^j*C(k-1,j)*(k-j)^(n-k) for k>0, T(n,0) = A000007(n).

T(n,k) = (k-1)! * A136011(n,k) for n, k >= 1.

Sum_{j>=0} T(n+j,j) = A076726(n) = 2*A000670(n) = A000629(n) + A000007(n).

EXAMPLE

T(5,1) = 1: 12345.

T(5,2) = 7: 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345.

T(5,3) = 2: 124|3|5, 12|34|5.

T(7,4) = 6: 1246|3|5|7, 124|36|5|7, 124|3|56|7, 126|34|5|7, 12|346|5|7, 12|34|56|7.

T(9,5) = 24: 12468|3|5|7|9, 1246|38|5|7|9, 1246|3|58|7|9, 1246|3|5|78|9, 1248|36|5|7|9, 124|368|5|7|9, 124|36|58|7|9, 124|36|5|78|9, 1248|3|56|7|9, 124|38|56|7|9, 124|3|568|7|9, 124|3|56|78|9, 1268|34|5|7|9, 126|348|5|7|9, 126|34|58|7|9, 126|34|5|78|9, 128|346|5|7|9, 12|3468|5|7|9, 12|346|58|7|9, 12|346|5|78|9, 128|34|56|7|9, 12|348|56|7|9, 12|34|568|7|9, 12|34|56|78|9.

Triangle T(n,k) begins:

  1;

  0, 1;

  0, 1;

  0, 1,    1;

  0, 1,    3;

  0, 1,    7,     2;

  0, 1,   15,    12;

  0, 1,   31,    50,     6;

  0, 1,   63,   180,    60;

  0, 1,  127,   602,   390,    24;

  0, 1,  255,  1932,  2100,   360;

  0, 1,  511,  6050, 10206,  3360,  120;

  0, 1, 1023, 18660, 46620, 25200, 2520;

  ...

MAPLE

b:= proc(n, m, t) option remember; `if`(n=0, x^m, add(

      b(n-1, max(m, j), `if`(j>m, 1, 0)), j=1..m+1-t))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):

seq(T(n), n=0..14);

# second Maple program:

T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), (k-1)!*Stirling2(n-k+1, k)):

seq(seq(T(n, k), k=0..ceil(n/2)), n=0..14);

# third Maple program:

T:= proc(n, k) option remember; `if`(k<2, `if`(n=0 xor k=0, 0, 1),

      `if`(k>ceil(n/2), 0, add((k-j)*T(n-1-j, k-j), j=0..1)))

    end:

seq(seq(T(n, k), k=0..ceil(n/2)), n=0..14);

MATHEMATICA

T[n_, k_] := T[n, k] = If[k < 2, If[Xor[n == 0, k == 0], 0, 1],

     If[k > Ceiling[n/2], 0, Sum[(k-j) T[n-1-j, k-j], {j, 0, 1}]]];

Table[Table[T[n, k], {k, 0, Ceiling[n/2]}], {n, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, Mar 08 2021, after third Maple program *)

CROSSREFS

Columns k=0-11 give (offsets may differ): A000007, A057427, A168604, A028243, A028244, A028245, A032180, A228909, A228910, A228911, A228912, A228913.

Row sums give A229046(n-1) for n>0.

T(2n+1,n+1) gives A000142.

T(2n,n) gives A001710(n+1).

Cf. A000629, A000670, A008277, A028246, A048993, A076726, A110654, A136011, A173018.

Sequence in context: A216802 A297786 A214407 * A137680 A248722 A201663

Adjacent sequences:  A298665 A298666 A298667 * A298669 A298670 A298671

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Jan 24 2018

STATUS

approved

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Last modified June 12 16:18 EDT 2021. Contains 344959 sequences. (Running on oeis4.)