A210391 -id:A210391 - cp's OEIS frontend

A139594 Number of different n X n symmetric matrices with nonnegative entries summing to 4. Also number of symmetric oriented graphs with 4 arcs on n points.

0, 1, 9, 39, 116, 275, 561, 1029, 1744, 2781, 4225, 6171, 8724, 11999, 16121, 21225, 27456, 34969, 43929, 54511, 66900, 81291, 97889, 116909, 138576, 163125, 190801, 221859, 256564, 295191, 338025, 385361, 437504, 494769, 557481, 625975, 700596

Offset: 0

Marc A. A. van Leeuwen, Jun 12 2008

a(n) is also the number of semistandard Young tableaux over all partitions of 4 with maximal element <= n. - Alois P. Heinz, Mar 22 2012

Starting from 1 the partial sums give A244864. - J. M. Bergot, Sep 17 2016

			From _Michael B. Porter_, Sep 18 2016: (Start)
The nine 2 X 2 matrices summing to 4 are:
4 0  3 0  2 0  1 0  0 0  2 1  1 1  0 1  0 2
0 0  0 1  0 2  0 3  0 4  1 0  1 1  1 2  2 0
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

For 3 in place of 4 this gives A005900.
Row n=4 of A210391. - Alois P. Heinz, Mar 22 2012
Partial sums of A063489.

dd := proc(n,m) coeftayl(1/((1-X)^m*(1-X^2)^binomial(m,2)),X=0,n); seq(dd(4,m),m=0..N);

gf[k_] := 1/((1-x)^k (1-x^2)^(k(k-1)/2));
T[n_, k_] := SeriesCoefficient[gf[k], {x, 0, n}];
a[k_] := T[4, k];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 07 2020 *)

a(n) = coefficient of x^4 in 1/((1-x)^n * (1-x^2)^binomial(n,2)).

a(n) = (n^2*(7+5*n^2))/12. G.f.: x*(1+x)*(1+3*x+x^2)/(1-x)^5. [Colin Barker, Mar 18 2012]

A181480 a(n) has generating function 1/((1-x)^k(1-x^2)^(k(k-1)/2)) for k=8.

Original entry on oeis.org

1, 8, 64, 344, 1744, 7400, 29632, 106808, 366088, 1168008, 3570240, 10347864, 28915056, 77493096, 201249216, 505130808, 1233655332, 2927916264, 6784208704, 15338678264, 33950726992, 73557910088, 156378379456, 326236930136, 669101503096, 1349416997864

Offset: 0

Wouter Meeussen, Oct 24 2010

a(n-1,k) is conjectured to also be the count of monomials (or terms) in the Schur polynomials of k variables and degree n, summed over all partitions of n in at most k parts (zero-padded to length k).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Schur Polynomial

For k=2 (two variables): A002620, k=3: A038163, k=4: A054498, k=5: A181477, k=6: A181478, k=7: A181479.

Column k=8 of A210391. - Alois P. Heinz, Mar 22 2012

CoefficientList[Series[1/(1-x)^8/(1-x^2)^28,{x,0,25}],x]

A210427 Number of semistandard Young tableaux over all partitions of 5 with maximal element <= n.

Original entry on oeis.org

0, 1, 12, 69, 260, 751, 1812, 3843, 7400, 13221, 22252, 35673, 54924, 81731, 118132, 166503, 229584, 310505, 412812, 540493, 698004, 890295, 1122836, 1401643, 1733304, 2125005, 2584556, 3120417, 3741724, 4458315, 5280756, 6220367, 7289248, 8500305, 9867276

Offset: 0

Alois P. Heinz, Mar 21 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Row n=5 of A210391.

a:= n-> n*(12+(35+13*n^2)*n^2)/60:
seq(a(n), n=0..40);

LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,12,69,260,751},40] (* Harvey P. Dale, Sep 20 2020 *)

a(n) = n*(12+(35+13*n^2)*n^2)/60.

G.f.: x*(12*x^2+6*x^3+x^4+1+6*x)/(x-1)^6.

A210428 Number of semistandard Young tableaux over all partitions of 6 with maximal element <= n.

Original entry on oeis.org

0, 1, 16, 119, 560, 1955, 5552, 13573, 29632, 59229, 110320, 193963, 325040, 523055, 813008, 1226345, 1801984, 2587417, 3639888, 5027647, 6831280, 9145115, 12078704, 15758381, 20328896, 25955125, 32823856, 41145651, 51156784, 63121255, 77332880, 94117457

Offset: 0

Alois P. Heinz, Mar 21 2012

a(n) is the number of semistandard Young tableaux over all partitions of 6 with maximal element <= n.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Row n=6 of A210391.

a:= n-> n^2*(76+(85+19*n^2)*n^2)/180:
seq(a(n), n=0..40);

a(n) = n^2*(76+(85+19*n^2)*n^2)/180.

G.f.: -x*(x+1)*(x^4+8*x^3+20*x^2+8*x+1)/(x-1)^7.

A210429 Number of semistandard Young tableaux over all partitions of 7 with maximal element <= n.

Original entry on oeis.org

0, 1, 20, 189, 1100, 4615, 15372, 43219, 106808, 238581, 491380, 946913, 1726308, 3002987, 5018092, 8098695, 12679024, 19324937, 28761876, 41906533, 59902460, 84159855, 116399756, 158702875, 213563304, 283947325, 373357556, 485902665, 626372884, 800321555

Offset: 0

Alois P. Heinz, Mar 21 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Row n=7 of A210391.

a:= n-> n*(180+(637+(385+58*n^2)*n^2)*n^2)/1260:
seq(a(n), n=0..40);

CoefficientList[Series[x (x+1)^2(x^4+10x^3+36x^2+10x+1)/(x-1)^8,{x,0,40}],x] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,1,20,189,1100,4615,15372,43219},40] (* Harvey P. Dale, Jan 29 2023 *)

a(n) = n*(180+(637+(385+58*n^2)*n^2)*n^2)/1260.

G.f.: x*(x+1)^2*(x^4+10*x^3+36*x^2+10*x+1)/(x-1)^8.

A210430 Number of semistandard Young tableaux over all partitions of 8 with maximal element <= n.

Original entry on oeis.org

0, 1, 25, 294, 2090, 10460, 40677, 131131, 366088, 912519, 2075965, 4381168, 8683962, 16321682, 29310113, 50595765, 84373024, 136476493, 214859601, 330171322, 496443610, 731902920, 1059919949, 1510112495, 2119617096, 2934545875, 4011645781, 5420178180

Offset: 0

Alois P. Heinz, Mar 21 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Row n=8 of A210391.

a:= n-> n^2*(n^2+3)*(1124+(1205+191*n^2)*n^2)/10080:
seq(a(n), n=0..40);

a(n) = n^2*(n^2+3)*(1124+(1205+191*n^2)*n^2)/10080.

G.f.: -x*(x+1)*(x^6+15*x^5+90*x^4+170*x^3+90*x^2+15*x+1)/(x-1)^9.

A210431 Number of semistandard Young tableaux over all partitions of 9 with maximal element <= n.

Original entry on oeis.org

0, 1, 30, 434, 3740, 22220, 100562, 370909, 1168008, 3245311, 8148590, 18821968, 40542228, 82300842, 158779362, 293092635, 520505744, 893364637, 1487517086, 2410539918, 3812130380, 5897064040, 8941168786, 13310814265, 19486468504, 28090928475, 39922889006

Offset: 0

Alois P. Heinz, Mar 21 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Row n=9 of A210391.

a:= n-> n*(10080+(40484+(31395+(8106+655*n^2)*n^2)*n^2)*n^2)/90720:
seq(a(n), n=0..40);

a(n) = n*(10080+(40484+(31395+(8106+655*n^2)*n^2)*n^2)*n^2)/90720.

G.f.: x*(x^8+20*x^7+179*x^6+630*x^5+960*x^4+630*x^3+179*x^2+20*x+1) / (x-1)^10.

A210432 Number of semistandard Young tableaux over all partitions of 10 with maximal element <= n.

Original entry on oeis.org

0, 1, 36, 630, 6512, 45628, 239316, 1007083, 3570240, 11042199, 30569012, 77221232, 180646896, 395884866, 820217412, 1618520277, 3060257024, 5572071725, 9810869508, 16763347626, 27879160048, 45246275592, 71818632820, 111707913791, 170553162816, 255984075075

Offset: 0

Alois P. Heinz, Mar 21 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Row n=10 of A210391.

a:= n-> n^2* (126432+ (206020+ (101031+ (18930+ 1187*n^2) *n^2) *n^2) *n^2)/ 453600:
seq(a(n), n=0..40);

a(n) = n^2*(126432+(206020+(101031+(18930+1187*n^2)*n^2)*n^2)*n^2)/453600.

G.f.: -x*(x+1)*(x^8 +24*x^7 +265*x^6 +1132*x^5 +1904*x^4 +1132*x^3 +265*x^2 +24*x+1) / (x-1)^11.

A262030 Partition array in Abramowitz-Stegun order: Schur functions evaluated at 1.

Original entry on oeis.org

1, 3, 1, 10, 8, 1, 35, 45, 20, 15, 1, 126, 224, 175, 126, 75, 24, 1, 462, 1050, 1134, 490, 840, 896, 175, 280, 189, 35, 1, 1716, 4752, 6468, 4704, 4950, 7350, 3528, 2646, 2400, 2940, 784, 540, 392, 48, 1, 6435, 21021, 34320, 33264, 13860, 27027, 50688, 41580, 25872, 15876, 17325, 29700, 15120, 14700, 1764, 5775, 7680, 2352, 945, 720, 63, 1

Offset: 1

Wolfdieter Lang, Oct 15 2015

The length of row n >= 1 of this irregular triangle is A000041(n) (partition numbers).
The Abramowitz-Stegun (A-St) order of the partitions is used.
For Schur functions (or polynomials) s^(lambda) = s(lambda(n, k);x[1], ..., x[n]) defined for the k-th partition of n (here in A-St order) see, e.g., the Macdonald reference, ch. I, 3. S-functions, p. 23, and Wikipedia reference. For the combinatorial interpretation of Schur functions see the Stanley reference.
The partition lambda(n,k) has m = m(n,k) nonincreasing parts lamda_j, j = 1..m, (the reverse of the partitions given in A-St) and n-m 0's are appended to obtain a length n partition. E.g., lambda(4, 3) = (2, 2, 0, 0) with m = 2.
The Schur function s(lambda(n, k),1,...,1) (n 1's) gives the number of semistandard Young tableaux (SSYT) for the Young (Ferrers) diagram of lambda(n, k) (forgetting about trailing 0's) with the box numbers taken out of the set {1, 2, ..., n} where the rows increase weakly and the columns increase strictly. See the Stanley reference pp. 309 and 310, and the example below.
The sum of the row numbers give A209673: 1, 4, 19, 116, 751, 5552, 43219, 366088, ...
Conjecture: The sum of the squares of row numbers give A054688: 1, 10, 165, 3876, ... = binomial(n^2+n-1, n). - Wouter Meeussen, Sep 25 2016

			The irregular triangle begins (commas separate entries for partitions of like numbers of parts in A-St order):
n\k  1     2    3    4    5   6   7    8   9 10 11
1:   1
2:   3,    1
3:  10,    8,   1
4:  35,   45   20,  15,   1
5: 126,  224  175, 126   75, 24,  1
6: 462, 1050 1134  490, 840 896 175, 280 189,35, 1
...
Row 7: 1716, 4752 6468 4704, 4950 7350 3528 2646, 2400 2940 784, 540 392, 48, 1;
Row 8: 6435, 21021 34320 33264 13860, 27027 50688 41580 25872 15876, 17325 29700 15120 14700 1764, 5775 7680 2352, 945 720, 63, 1.
...
n = 4, k = 4: lambda(4, 4) = (2,1,1,0) (m=3), SSYT (we use semicolons to separate the three rows): [1,1;2;3], [1,1;2;4], [1,1;3;4],
  [1,2;2;3], [1,2;2;4], [1,2;3;4],
  [1,3;2;3], [1,3;2;4], [1,3;3;4],
  [1,4;2;3], [1,4;2;4], [1,4;3;4],
  [2,2;3;4], [2,3;3;4], [2,4;3;4], hence a(4, 4) = 15. The three tableaux with distinct numbers are standard Young tableaux and give A117506(4, 4) = 3.

Macdonald, I. G., Symmetric functions and Hall polynomials, Oxford University Press, 1979.
Stanley, R. P., Enumerative Combinatorics, Vol. 2, Cambridge University Press 1999, sect. 7.30, pp. 308-316.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Wikipedia, Schur functions.

Cf. A054688, A117506, A209673, A210391 (in the diagonal).

a(n, k) = Det_{i,j=1..n} x[i]^(lambda_j + n-j) / Det_{i,j=1..n} x[i]^(n-j), evaluated at x[i] = 1 for i = 1..n (after division). The denominator is the Vandermonde determinant, the numerator an alternant. See, e.g., the Macdonald reference p. 24.

A296560 Number of normal semistandard Young tableaux whose shape is the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 4, 6, 6, 1, 12, 1, 8, 16, 8, 1, 28, 1, 24, 30, 10, 1, 32, 22, 12, 44, 40, 1, 96, 1, 16, 48, 14, 68, 96, 1, 16, 70, 80, 1, 220, 1, 60, 204, 18, 1, 80, 90, 168, 96, 84, 1, 224, 146, 160, 126, 20, 1, 400, 1, 22, 584, 32, 264, 416, 1, 112, 160

Offset: 1

Gus Wiseman, Feb 15 2018

nonn

A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A191714, A210391, A228125, A296150, A296188, A299202.

a[n_]:=If[n===1,1,Sum[a[n/q*Times@@Cases[FactorInteger[q],{p_,k_}:>If[p===2,1,NextPrime[p,-1]^k]]],{q,Rest[Divisors[n]]}]];
Array[a,100]

cp's OEIS Frontend

A139594 Number of different n X n symmetric matrices with nonnegative entries summing to 4. Also number of symmetric oriented graphs with 4 arcs on n points.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A181480 a(n) has generating function 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)) for k=8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A210427 Number of semistandard Young tableaux over all partitions of 5 with maximal element <= n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A210428 Number of semistandard Young tableaux over all partitions of 6 with maximal element <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A210429 Number of semistandard Young tableaux over all partitions of 7 with maximal element <= n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A210430 Number of semistandard Young tableaux over all partitions of 8 with maximal element <= n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A210431 Number of semistandard Young tableaux over all partitions of 9 with maximal element <= n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A210432 Number of semistandard Young tableaux over all partitions of 10 with maximal element <= n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A262030 Partition array in Abramowitz-Stegun order: Schur functions evaluated at 1.

A181480 a(n) has generating function 1/((1-x)^k(1-x^2)^(k(k-1)/2)) for k=8.