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%I #27 Aug 22 2021 13:45:47
%S 1,1,2,3,6,12,27,63,154,398,1055,2970,8503,25651,78483,250487,811802,
%T 2723130,9295483,32653552,116866283,428464743,1600474365,6102119282,
%U 23690388631,93631999867,376561553417,1538997717423,6395852269479,26978392034357,115628083386280,502520979828775
%N Number of standard shifted tableaux with n entries.
%C Number of ballot sequences (see A000085) where the number of occurrences of k in any prefix is strictly greater than the number of occurrences of k+1. - _Joerg Arndt_, May 21 2016
%D D. E. Knuth, The Art of Computer Programming, Vol. 3 (Sorting and searching), page 71, Section 5.1.4, Exercise 21 (page 67 in the second edition).
%H Joerg Arndt, <a href="/A061343/b061343.txt">Table of n, a(n) for n = 1..101</a>
%H Joerg Arndt, <a href="/A061343/a061343.gp.txt">PARI/GP script</a> to compute terms.
%H R. Srinivasan, <a href="http://www.jstor.org/stable/2312782">On a theorem of Thrall in combinatorial analysis</a>, The American Mathematical Monthly, 70(1), 1963, pp. 41-44.
%H R. M. Thrall, <a href="http://projecteuclid.org/euclid.mmj/1028989731">A combinatorial problem</a>, Michigan Math. J. 1, (1952), 81-88.
%F a(n) is the sum over all partitions into distinct parts of Thrall's formula (4) on page 83, see the PARI script arndt-A061343.gp. - _Joerg Arndt_, May 09 2013
%e From _Joerg Arndt_, May 21 2016: (Start)
%e The a(7) = 27 tableaux correspond to the following ballot sequences (dots denote zeros).
%e ##: ballot sequence partition
%e 01: [ . . . . . . . ] [ 7 . . . . . . ]
%e 02: [ . . . . . . 1 ] [ 6 1 . . . . . ]
%e 03: [ . . . . . 1 . ] [ 6 1 . . . . . ]
%e 04: [ . . . . . 1 1 ] [ 5 2 . . . . . ]
%e 05: [ . . . . 1 . . ] [ 6 1 . . . . . ]
%e 06: [ . . . . 1 . 1 ] [ 5 2 . . . . . ]
%e 07: [ . . . . 1 1 . ] [ 5 2 . . . . . ]
%e 08: [ . . . . 1 1 1 ] [ 4 3 . . . . . ]
%e 09: [ . . . . 1 1 2 ] [ 4 2 1 . . . . ]
%e 10: [ . . . 1 . . . ] [ 6 1 . . . . . ]
%e 11: [ . . . 1 . . 1 ] [ 5 2 . . . . . ]
%e 12: [ . . . 1 . 1 . ] [ 5 2 . . . . . ]
%e 13: [ . . . 1 . 1 1 ] [ 4 3 . . . . . ]
%e 14: [ . . . 1 . 1 2 ] [ 4 2 1 . . . . ]
%e 15: [ . . . 1 1 . . ] [ 5 2 . . . . . ]
%e 16: [ . . . 1 1 . 1 ] [ 4 3 . . . . . ]
%e 17: [ . . . 1 1 . 2 ] [ 4 2 1 . . . . ]
%e 18: [ . . . 1 1 2 . ] [ 4 2 1 . . . . ]
%e 19: [ . . 1 . . . . ] [ 6 1 . . . . . ]
%e 20: [ . . 1 . . . 1 ] [ 5 2 . . . . . ]
%e 21: [ . . 1 . . 1 . ] [ 5 2 . . . . . ]
%e 22: [ . . 1 . . 1 1 ] [ 4 3 . . . . . ]
%e 23: [ . . 1 . . 1 2 ] [ 4 2 1 . . . . ]
%e 24: [ . . 1 . 1 . . ] [ 5 2 . . . . . ]
%e 25: [ . . 1 . 1 . 1 ] [ 4 3 . . . . . ]
%e 26: [ . . 1 . 1 . 2 ] [ 4 2 1 . . . . ]
%e 27: [ . . 1 . 1 2 . ] [ 4 2 1 . . . . ]
%e (End)
%Y Cf. A000085, A003121 (strict ballot sequences with partition [j, j-1, ..., 3, 2, 1]).
%K nonn,nice
%O 1,3
%A V. Reiner and D. White (reiner(AT)math.umn.edu), Jun 07 2001
%E More terms from _Joerg Arndt_, May 08 2013
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