This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A061343 #27 Aug 22 2021 13:45:47 %S A061343 1,1,2,3,6,12,27,63,154,398,1055,2970,8503,25651,78483,250487,811802, %T A061343 2723130,9295483,32653552,116866283,428464743,1600474365,6102119282, %U A061343 23690388631,93631999867,376561553417,1538997717423,6395852269479,26978392034357,115628083386280,502520979828775 %N A061343 Number of standard shifted tableaux with n entries. %C A061343 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 A061343 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 A061343 Joerg Arndt, <a href="/A061343/b061343.txt">Table of n, a(n) for n = 1..101</a> %H A061343 Joerg Arndt, <a href="/A061343/a061343.gp.txt">PARI/GP script</a> to compute terms. %H A061343 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 A061343 R. M. Thrall, <a href="http://projecteuclid.org/euclid.mmj/1028989731">A combinatorial problem</a>, Michigan Math. J. 1, (1952), 81-88. %F A061343 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 A061343 From _Joerg Arndt_, May 21 2016: (Start) %e A061343 The a(7) = 27 tableaux correspond to the following ballot sequences (dots denote zeros). %e A061343 ##: ballot sequence partition %e A061343 01: [ . . . . . . . ] [ 7 . . . . . . ] %e A061343 02: [ . . . . . . 1 ] [ 6 1 . . . . . ] %e A061343 03: [ . . . . . 1 . ] [ 6 1 . . . . . ] %e A061343 04: [ . . . . . 1 1 ] [ 5 2 . . . . . ] %e A061343 05: [ . . . . 1 . . ] [ 6 1 . . . . . ] %e A061343 06: [ . . . . 1 . 1 ] [ 5 2 . . . . . ] %e A061343 07: [ . . . . 1 1 . ] [ 5 2 . . . . . ] %e A061343 08: [ . . . . 1 1 1 ] [ 4 3 . . . . . ] %e A061343 09: [ . . . . 1 1 2 ] [ 4 2 1 . . . . ] %e A061343 10: [ . . . 1 . . . ] [ 6 1 . . . . . ] %e A061343 11: [ . . . 1 . . 1 ] [ 5 2 . . . . . ] %e A061343 12: [ . . . 1 . 1 . ] [ 5 2 . . . . . ] %e A061343 13: [ . . . 1 . 1 1 ] [ 4 3 . . . . . ] %e A061343 14: [ . . . 1 . 1 2 ] [ 4 2 1 . . . . ] %e A061343 15: [ . . . 1 1 . . ] [ 5 2 . . . . . ] %e A061343 16: [ . . . 1 1 . 1 ] [ 4 3 . . . . . ] %e A061343 17: [ . . . 1 1 . 2 ] [ 4 2 1 . . . . ] %e A061343 18: [ . . . 1 1 2 . ] [ 4 2 1 . . . . ] %e A061343 19: [ . . 1 . . . . ] [ 6 1 . . . . . ] %e A061343 20: [ . . 1 . . . 1 ] [ 5 2 . . . . . ] %e A061343 21: [ . . 1 . . 1 . ] [ 5 2 . . . . . ] %e A061343 22: [ . . 1 . . 1 1 ] [ 4 3 . . . . . ] %e A061343 23: [ . . 1 . . 1 2 ] [ 4 2 1 . . . . ] %e A061343 24: [ . . 1 . 1 . . ] [ 5 2 . . . . . ] %e A061343 25: [ . . 1 . 1 . 1 ] [ 4 3 . . . . . ] %e A061343 26: [ . . 1 . 1 . 2 ] [ 4 2 1 . . . . ] %e A061343 27: [ . . 1 . 1 2 . ] [ 4 2 1 . . . . ] %e A061343 (End) %Y A061343 Cf. A000085, A003121 (strict ballot sequences with partition [j, j-1, ..., 3, 2, 1]). %K A061343 nonn,nice %O A061343 1,3 %A A061343 V. Reiner and D. White (reiner(AT)math.umn.edu), Jun 07 2001 %E A061343 More terms from _Joerg Arndt_, May 08 2013