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A275787 Number of cells in the two-sided Coxeter complex of type B_n.

1, 5, 41, 509, 8469, 176217, 4400325, 128203049, 4268957449, 159922273421, 6656731517249, 304797275277365, 15224868078068845, 823874409422614577, 48012621942105876301, 2997884066292303095889, 199666128081901473290833, 14129411123649333432720277, 1058688691179737704258634521, 83732563305101190468369022317, 6971039973751002759723517967941

Offset: 1

Kyle Petersen, Aug 09 2016

nonn

a(n) is the number of nonnegative integer matrices with sum of entries equal to 2*n-2 (or 2*n-1), no zero rows or columns, which are centrally symmetric. - Ludovic Schwob, Feb 17 2024

			The a(2) = 5 matrices whose sum of entries is equal to 2:
  [2] [1 1]
.
  [1] [1 0] [0 1]
  [1] [0 1] [1 0]

Evgeniy Krasko, Counting Unlabelled Chord Diagrams of Maximal Genus, arXiv:1709.00796 [math.CO], 2017.
T. K. Petersen, A two-sided analogue of the Coxeter complex, arXiv:1607.00086 [math.CO], 2016.

Cf. A120733 gives the number of cells for type A_n.

B:=proc(n) local f;
option remember;
if n=1 then 1+s*t;
elif n>1 then
f:=B(n-1);
RETURN(simplify( (2*n*s*t-s*t+1)*f+(2*s*t*(1-s)+s/n*(1-s)*(1-t))*diff(f,s) + (2*s*t*(1-t)+t/n*(1-s)*(1-t))*diff(f,t) + 2/n*s*t*(1-s)*(1-t)*diff( diff(f,s),t) ));
fi;
end:
seq(eval(eval(subs(s=x/(1+x),t=y/(1+y), B(n))*(1+x)^n*(1+y)^n,y=1),x=1), n=1..30);

B[n_] := B[n] = Which[n == 1, 1 + s*t, n > 1, f = B[n - 1]; Return[ Simplify[ (2*n*s*t - s*t + 1)*f + (2*s*t*(1 - s) + s/n*(1 - s)*(1 - t))*D[f, s] + (2*s*t*(1 - t) + t/n*(1 - s)*(1 - t))*D[f, t] + 2/n*s*t*(1 - s)*(1 - t)*D[ D[f, s], t]]]];
Join[{1}, Table[bn = ((B[n] /. {s -> x/(1 + x), t -> y/(1 + y)})*(1 + x)^n*(1 + y)^n /. {y -> 1, x -> 1}); Print[bn]; bn, {n, 1, 20}]] (* Jean-François Alcover, Nov 27 2017, from Maple *)

A260700 Number of distinct parabolic double cosets of the symmetric group S_n.

Original entry on oeis.org

1, 3, 19, 167, 1791, 22715, 334031, 5597524, 105351108, 2200768698, 50533675542, 1265155704413, 34300156146805, 1001152439025205, 31301382564128969, 1043692244938401836, 36969440518414369896, 1386377072447199902576, 54872494774746771827248, 2285943548113541477123970

Offset: 1

Kyle Petersen, Nov 16 2015

nonn

This is closely related to the number of contingency tables on n elements (see A120733), but many contingency tables correspond to the same parabolic double coset, e.g., for n=2, there are 5 contingency tables, but only 3 distinct cosets.

			For n=2, there are three parabolic double cosets: {12}, {21}, and {12, 21}.

Thomas Browning, Table of n, a(n) for n = 1..400
Sara Billey, Matjaz Konvalinka, T. Kyle Petersen, William Slofstra, and Bridget Tenner, Parabolic double cosets in Coxeter groups, Discrete Mathematics and Theoretical Computer Science, Submitted, 2016.
Sara Billey, Matjaz Konvalinka, T. Kyle Petersen, William Slofstra, and Bridget Tenner, Parabolic double cosets in Coxeter groups, Electron. J. Combin., Volume 25, Issue 1 (2018) P1.23.
Thomas Browning, Counting Parabolic Double Cosets in Symmetric Groups, arXiv:2010.13256 [math.CO], 2020.
Masato Kobayashi, Construction of double coset system of a Coxeter group and its applications to Bruhat graphs, arXiv:1907.11801 [math.CO], 2019.

Cf. A120733.

a(n) is asymptotic to n! / (2^(log(2)/2 + 2) * log(2)^(2*n + 2)). [Conjectured Vaclav Kotesovec Sep 08 2020, proved Thomas Browning Oct 26 2020]

More terms from Thomas Browning, Sep 07 2020

A214418 Sum of absolute values of the character table of the symmetric group S_n.

Original entry on oeis.org

1, 4, 9, 27, 61, 185, 469, 1428, 4292, 14456, 50040, 186525, 724023, 2952085, 12476584, 54670003, 246823085, 1148176939, 5479962138, 26831059535, 134411101756, 688922857792, 3604792968605, 19250556430137, 104789182807326, 581220024906812, 3280986439096740

Offset: 1

Kyle Petersen, Jul 17 2012

nonn

			The character table of S_3 is / 1 1 1 / 2 0 -1 / 1 -1 1 / and so a(3)=9.

Eric M. Schmidt, Table of n, a(n) for n = 1..35
T. Kyle Petersen and Bridget Eileen Tenner, How to write a permutation as a product of involutions (and why you might care), arXiv:1202.5319 [math.CO], 2012.

Row sums of A214430. Compare with A082733. Conjecturally dominated by A164342.

A214418 := n -> Sum(Irr(CharacterTable("Symmetric", n)), chi -> Sum(chi, x->AbsoluteValue(x))); # Eric M. Schmidt, Jul 18 2012, revised Sep 05 2012

P:=combinat[partition](n):
add(add(abs(combinat[Chi](l, m)), l in P), m in P);

A214430 Triangle read by rows, where T(n,m) is sum of the absolute values of the m-th column (in lexicographic ordering) in the character table of S_n.

Original entry on oeis.org

1, 2, 2, 4, 2, 3, 10, 4, 6, 3, 4, 26, 8, 6, 6, 6, 4, 5, 76, 20, 12, 20, 12, 6, 12, 8, 8, 5, 6, 232, 52, 24, 20, 30, 12, 18, 12, 16, 8, 12, 10, 10, 6, 7, 764, 148, 52, 36, 76, 78, 24, 18, 24, 24, 36, 12, 20, 12, 20, 20, 10, 15, 12, 12, 7, 8, 2620, 460, 148, 76, 76, 208, 56, 32, 56, 40, 24, 54, 100, 28, 20, 20, 20, 20, 50

Offset: 1

Kyle Petersen, Jul 17 2012

Ordering on partitions is lexicographic, where partitions themselves are written in decreasing order, e.g., for n=5, the order is [1,1,1,1,1] < [2,1,1,1] < [2,2,1] < [3,1,1] < [3,2] < [4,1] < [5].

			The character table for S_3 is / 1 1 1 / 2 0 -1 / 1 -1 1 / and so T(3,1)=4, T(3,2)=2, and T(3,3)=3.
Displayed as a triangle:
1
2, 2
4, 2, 3
10, 4, 6, 3, 4
26, 8, 6, 6, 6, 4, 5
76, 20, 12, 20, 12, 6, 12, 8, 8, 5, 6
232, 52, 24, 20, 30, 12, 18, 12, 16, 8, 12, 10, 10, 6, 7
764, 148, 52, 36, 76, 78, 24, 18, 24, 24, 36, 12, 20, 12, 20, 20, 10, 15, 12, 12, 7, 8

Kyle Petersen, Table of n, a(n) for n = 1..138
T. Kyle Petersen and Bridget Eileen Tenner, How to write a permutation as a product of involutions (and why you might care), arXiv:1202.5319 [math.CO], 2012.

Equal to A164341 for n<=7, row sums given in A214418. First column, corresponding to partition [1,1,...,1], is given by A000085.

#For row n, we have the following.
P:=combinat[partition](n):
seq(add(abs(combinat[Chi](l, m)), l in P), m in P);

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A275787 Number of cells in the two-sided Coxeter complex of type B_n.

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A260700 Number of distinct parabolic double cosets of the symmetric group S_n.

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A214418 Sum of absolute values of the character table of the symmetric group S_n.

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Maple

A214430 Triangle read by rows, where T(n,m) is sum of the absolute values of the m-th column (in lexicographic ordering) in the character table of S_n.

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Author

Keywords

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Examples

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Maple