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A061343 Number of standard shifted tableaux with n entries. 1
1, 1, 2, 3, 6, 12, 27, 63, 154, 398, 1055, 2970, 8503, 25651, 78483, 250487, 811802, 2723130, 9295483, 32653552, 116866283, 428464743, 1600474365, 6102119282, 23690388631, 93631999867, 376561553417, 1538997717423, 6395852269479, 26978392034357, 115628083386280, 502520979828775 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

REFERENCES

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).

LINKS

Joerg Arndt, Table of n, a(n) for n = 1..101

Joerg Arndt, Pari/GP script to compute terms.

R. Srinivasan, On a theorem of Thrall in combinatorial analysis, The American Mathematical Monthly, 70(1), 1963, pp. 41-44.

R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88.

FORMULA

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]

EXAMPLE

From Joerg Arndt, May 21 2016: (Start)

The a(7) = 27 tableaux correspond to the following ballot sequences (dots denote zeros).

##:     ballot sequence          partition

01:    [ . . . . . . . ]       [ 7 . . . . . . ]

02:    [ . . . . . . 1 ]       [ 6 1 . . . . . ]

03:    [ . . . . . 1 . ]       [ 6 1 . . . . . ]

04:    [ . . . . . 1 1 ]       [ 5 2 . . . . . ]

05:    [ . . . . 1 . . ]       [ 6 1 . . . . . ]

06:    [ . . . . 1 . 1 ]       [ 5 2 . . . . . ]

07:    [ . . . . 1 1 . ]       [ 5 2 . . . . . ]

08:    [ . . . . 1 1 1 ]       [ 4 3 . . . . . ]

09:    [ . . . . 1 1 2 ]       [ 4 2 1 . . . . ]

10:    [ . . . 1 . . . ]       [ 6 1 . . . . . ]

11:    [ . . . 1 . . 1 ]       [ 5 2 . . . . . ]

12:    [ . . . 1 . 1 . ]       [ 5 2 . . . . . ]

13:    [ . . . 1 . 1 1 ]       [ 4 3 . . . . . ]

14:    [ . . . 1 . 1 2 ]       [ 4 2 1 . . . . ]

15:    [ . . . 1 1 . . ]       [ 5 2 . . . . . ]

16:    [ . . . 1 1 . 1 ]       [ 4 3 . . . . . ]

17:    [ . . . 1 1 . 2 ]       [ 4 2 1 . . . . ]

18:    [ . . . 1 1 2 . ]       [ 4 2 1 . . . . ]

19:    [ . . 1 . . . . ]       [ 6 1 . . . . . ]

20:    [ . . 1 . . . 1 ]       [ 5 2 . . . . . ]

21:    [ . . 1 . . 1 . ]       [ 5 2 . . . . . ]

22:    [ . . 1 . . 1 1 ]       [ 4 3 . . . . . ]

23:    [ . . 1 . . 1 2 ]       [ 4 2 1 . . . . ]

24:    [ . . 1 . 1 . . ]       [ 5 2 . . . . . ]

25:    [ . . 1 . 1 . 1 ]       [ 4 3 . . . . . ]

26:    [ . . 1 . 1 . 2 ]       [ 4 2 1 . . . . ]

27:    [ . . 1 . 1 2 . ]       [ 4 2 1 . . . . ]

(End)

CROSSREFS

Cf. A000085, A003121 (strict ballot sequences with partition [j, j-1, ..., 3, 2, 1]).

Sequence in context: A019525 A108915 A082395 * A057649 A104872 A006082

Adjacent sequences:  A061340 A061341 A061342 * A061344 A061345 A061346

KEYWORD

nonn,nice

AUTHOR

V. Reiner and D. White (reiner(AT)math.umn.edu), Jun 07 2001

EXTENSIONS

More terms by Joerg Arndt, May 08 2013

STATUS

approved

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Last modified March 22 22:17 EDT 2017. Contains 283901 sequences.