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A109062 Triangle read by rows: number of atomic set compositions of size n and length k (see description in A095989) 1 <= k <= n. 6
1, 1, 1, 1, 4, 3, 1, 11, 23, 13, 1, 26, 112, 158, 71, 1, 57, 446, 1170, 1241, 461, 1, 120, 1593, 6880, 12871, 10912, 3447, 1, 247, 5337, 35503, 103887, 150413, 106031, 29093, 1, 502, 17190, 168982, 724148, 1589266, 1872286, 1128218, 273343, 1, 1013, 54008 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Also the number of free generators and primitives of the quasi-symmetric functions in non-commuting variables 1; 1, 1; 1, 4, 3; 1, 11, 23, 13; 1, 26, 112, 158, 71. - Mike Zabrocki, Aug 06 2006

Triangle given by [1,0,2,0,3,0,4,0,5,...] DELTA [1,2,2,3,3,4,4,5,5,6,6,7,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 01 2007

LINKS

Table of n, a(n) for n=1..48.

N. Bergeron, M. Zabrocki, The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree, arXiv:math/0509265 [math.CO], 2005.

FORMULA

G.f.: 1-1/(1+Sum_{n>=1} Sum_{k=1..n} q^n*t^k*Stirling2(n,k)*k!).

EXAMPLE

Atomic set compositions a(1,1)=1: [{1}]; a(2,1)=1, a(2,2)=1: [{12}], [{2},{1}];

a(3,1)=1, a(3,2)=4, a(3,3)=3: [{123}], [{2},{13}], [{3}, {12}], [{23}, {1}], [{13},{2}], [{2},{3},{1}], [{3},{1},{2}], [{3},{2},{1}]

MAPLE

f:=(n, k)->coeff(coeff(series(1-1/(1+add(add(q^m*t^i*

    Stirling2(m, i)*i!, i=1..m), m=1..n)), q, n+1), q, n), t, k):

seq(seq(f(n, k), k=1..n), n=1..10);

CROSSREFS

Row sums are equal to A095989, a(n,n) = A003319, a(n,2) = A000295.

Cf. A095989, A059438, A074664, A087903, A008277, A019538.

Sequence in context: A157894 A172106 A128813 * A112493 A010305 A308326

Adjacent sequences:  A109059 A109060 A109061 * A109063 A109064 A109065

KEYWORD

nonn,tabl

AUTHOR

Mike Zabrocki, Aug 24 2005

STATUS

approved

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Last modified September 26 16:43 EDT 2020. Contains 337374 sequences. (Running on oeis4.)