A232439 -id:A232439 - cp's OEIS frontend

A225616 Number of tableaux of size n with major index (sum of descent set) equal to 1 mod n.

0, 1, 1, 2, 5, 12, 33, 94, 290, 949, 3245, 11666, 43731, 170748, 689957, 2887816, 12458783, 55406276, 253142181, 1187931688, 5712032811, 28131119950, 141645386201, 728841245442, 3827217750406, 20499431084637, 111876916513388, 621831333931916, 3516904353610571

Offset: 1

Wouter Meeussen, Aug 04 2013

nonn

Related to sum of characters of Lie(n).

Alois P. Heinz, Table of n, a(n) for n = 1..40
R. P. Stanley, Whitehouse

Cf. A232439.

descentset[t_?TableauQ]:=Sort[Cases[t,i_Integer /; Position[t,i+1][[1,1]] > Position[t,i][[1,1]], {2}]];
majorindex[t_?TableauQ]:=Tr[descentset[t]];
Table[Tr[Count[Tableaux[#],(q_/;Mod[majorindex[q],Tr[#]]==1) ]& /@ Partitions[n]],{n,13}]

a(n) = Sum_{k=0..n(n-1)/2, k mod n=1} A232439(n,k). - Alois P. Heinz, Sep 15 2014

Terms verified and more terms added, Joerg Arndt, Aug 04 2013
a(21)-a(24) from Alois P. Heinz, Aug 08 2013
a(25) from Alois P. Heinz, Aug 12 2013
a(26)-a(29) from Alois P. Heinz, Sep 15 2014

A247386 Sum of the major index over all standard Young tableaux with n cells.

Original entry on oeis.org

0, 0, 1, 6, 30, 130, 570, 2436, 10696, 47160, 213660, 981640, 4625016, 22171656, 108766840, 543350640, 2772404160, 14402353216, 76294507536, 411229473120, 2257073089120, 12595032754080, 71481175794336, 412117251151936, 2413922299708800, 14352066564086400

Offset: 0

Joerg Arndt and Alois P. Heinz, Sep 15 2014

nonn

Also sum of positions of all ascents in ballot sequences of length n, see example.

			The 10 ballot sequences of length 4 are:
##   [ ballot seq] ascent positions  major index
01:  [ 1 1 1 1 ]   (none)            0
02:  [ 1 1 1 2 ]   3                 3
03:  [ 1 1 2 1 ]   2                 2
04:  [ 1 1 2 2 ]   2                 2
05:  [ 1 1 2 3 ]   2 + 3             5
06:  [ 1 2 1 1 ]   1                 1
07:  [ 1 2 1 2 ]   1 + 3             4
08:  [ 1 2 1 3 ]   1 + 3             4
09:  [ 1 2 3 1 ]   1 + 2             3
10:  [ 1 2 3 4 ]   1 + 2 + 3         6
The sum of the major indexes is a(4) = 30.

Alois P. Heinz, Table of n, a(n) for n = 0..40
Wikipedia, Young tableau

Cf. A232439.

a(n) = Sum_{k=1..n*(n-1)/2} k*A232439(n,k).

cp's OEIS Frontend

A225616 Number of tableaux of size n with major index (sum of descent set) equal to 1 mod n.

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