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Search: id:A109062
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| A109062 |
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Triangle read by rows: number of atomic set compositions of size n and length k (see description in A095989) 1 <= k <= n. |
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+0
7
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| 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; listen)
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OFFSET
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1,5
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COMMENT
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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 (zabrocki(AT)mathstat.yorku.ca), 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 DELEHAM (kolotoko(AT)wanadoo.fr), Aug 01 2007
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REFERENCES
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N. Bergeron, M. Zabrocki, The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree, math.CO/0509265
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FORMULA
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G.f.: 1-1/(1+add(add(q^n*t^k*stirling2(n, k)*k!, k=1..n), n=1.. infinity)); stirling2(n, k) are the Stirling #'s of the second kind A008277
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EXAMPLE
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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}]
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MAPLE
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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);
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CROSSREFS
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Row sums are equal to A095989, a(n, n) = A003319, a(n, 2) = A000295
Cf. A095989, A059438, A074664, A087903, A008277, A019538.
Sequence in context: A109692 A157894 A128813 this_sequence A112493 A010305 A098234
Adjacent sequences: A109059 A109060 A109061 this_sequence A109063 A109064 A109065
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KEYWORD
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nonn,tabl
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AUTHOR
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Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 24 2005
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