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A317553
Sum of coefficients in the expansion of Sum_{y a composition of n} p(y) in terms of Schur functions, where p is power-sum symmetric functions.
3
1, 2, 5, 14, 39, 122, 387, 1328, 4675, 17414, 66743, 267234, 1100453, 4696414, 20580433, 92966560, 430394961, 2046068386, 9950230149, 49544789182, 251930150903, 1308655057210, 6931418152099, 37435337021328, 205874622937315, 1152718809407558, 6564213262312871
OFFSET
1,2
COMMENTS
From Hsin-Chieh Liao, Mar 25 2021: (Start)
Sum of coefficients of the Schur expansion of sum of Eulerian quasisymmetric functions Q_n,d over d from 0 to n-1.
a(n) is the number of marked tableaux with cells filled with 1,2,...,n. In Stembridge's paper, he gave a refinement of this number by the shape and the index of the tableaux. (End)
LINKS
Ludovic Schwob, Sage program
John Shareshian and Michelle. L. Wachs, Eulerian quasisymmetric functions, Advanced in Mathematics, 225(6) (2010), 2921-2966.
John R. Stembridge, Eulerian numbers, tableaux, and the Betti numbers of a toric variety, Discrete Mathematics, 99 (1992), 307-320.
EXAMPLE
We have p(4) + p(22) + 2 p(31) + 3 p(211) + p(1111) = 8 s(4) + 2 s(22) + 4 s(31), which has sum of coefficients a(4) = 14.
PROG
(Sage) # See Links - Ludovic Schwob, Sep 26 2023
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 14 2018
EXTENSIONS
a(11) onwards from Ludovic Schwob, Sep 26 2023
STATUS
approved