A006907 -id:A006907 - cp's OEIS frontend

A001255 Squares of partition numbers.

1, 1, 4, 9, 25, 49, 121, 225, 484, 900, 1764, 3136, 5929, 10201, 18225, 30976, 53361, 88209, 148225, 240100, 393129, 627264, 1004004, 1575025, 2480625, 3833764, 5934096, 9060100, 13823524, 20839225, 31404816, 46812964, 69705801, 102880449, 151536100

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A000041, A000290, A304990, A200089.

Cf. A133042, A054440.

a001255 = (^ 2) . a000041  -- Reinhard Zumkeller, Nov 15 2015

seq(numbpart(k)^2, k=0..33); # Zerinvary Lajos, Jun 06 2007

Table[ PartitionsP[n]^2, {n, 0, 33}] (* Ray Chandler, Nov 14 2005 *)

a(n) = numbpart(n)^2; \\ Michel Marcus, May 01 2021

a(n) = A000041(n)^2.
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (48*n^2). - Vaclav Kotesovec, Dec 01 2015
Sum_{n>=1} 1/a(n) = A200089. - Amiram Eldar, May 01 2021
a(n) = A006907(n) + A051748(n) + A051749(n). - R. J. Mathar, Mar 09 2022
a(n) = [(x*y)^n] Product_{k>=1} 1 / ((1 - x^k) * (1 - y^k)). - Ilya Gutkovskiy, Apr 24 2025

Extended by Ray Chandler, Nov 14 2005

A051748 Number of character table entries of the symmetric group S_n which are < 0.

Original entry on oeis.org

0, 1, 2, 7, 13, 34, 72, 137, 249, 524, 953, 1679, 3106, 5270, 9398, 15666, 26284, 43409, 72861, 115940, 189476, 297929, 476904, 743094, 1174624, 1782368, 2787256, 4196505, 6413986, 9645485, 14553197, 21483398, 32243250, 47165359, 69606943

Offset: 1

JOHN MCKAY (mckay(AT)cs.concordia.ca), Dec 08 1999

Cf. A006907, A051749.

A051748 := n -> Sum(Irr(CharacterTable("Symmetric", n)), chi -> Number(chi, x->x<0)); # Eric M. Schmidt, Jul 14 2012, revised Sep 05 2012

a[n_] := Count[FiniteGroupData[{"SymmetricGroup", n}, "CharacterTable"], ?Negative, 2]; Array[a, 10] (* _Jean-François Alcover, Apr 03 2017 *)

More terms from Eric M. Schmidt, Jul 14 2012

A051749 Number of character table entries of the symmetric group S_n which are > 0.

Original entry on oeis.org

1, 3, 6, 14, 26, 58, 98, 194, 344, 652, 1165, 2020, 3552, 6077, 10362, 17080, 28570, 46836, 77045, 122013, 198461, 310602, 494008, 767237, 1205391, 1828252, 2846995, 4277605, 6520106, 9795470, 14738493, 21750402, 32582580, 47614253, 70213289

Offset: 1

JOHN MCKAY (mckay(AT)cs.concordia.ca), Dec 08 1999

Cf. A006907, A051748.

A051749 := n -> Sum(Irr(CharacterTable("Symmetric", n)), chi -> Number(chi, x->x>0)); # Eric M. Schmidt, Jul 14 2012

a[n_] := Count[FiniteGroupData[{"SymmetricGroup", n}, "CharacterTable"], ?Positive, 2]; Array[a, 10] (* _Jean-François Alcover, Apr 03 2017 *)

A006907(n) + A051748(n) + a(n) = A001255(n). - R. J. Mathar, Mar 09 2022

More terms from Eric M. Schmidt, Jul 14 2012

A006908 Number of nonzero elements in the character table of the symmetric group S_n.

Original entry on oeis.org

1, 4, 8, 21, 39, 92, 170, 331, 593, 1176, 2118, 3699, 6658, 11347, 19760, 32746, 54854, 90245, 149906, 237953, 387937, 608531, 970912, 1510331, 2380015, 3610620, 5634251, 8474110, 12934092, 19440955, 29291690, 43233800, 64825830, 94779612, 139820232

Offset: 1

Simon Plouffe and N. J. A. Sloane

John McKay (email to N. J. A. Sloane, Apr 23 2013) observes that A061256 and A006908 coincide for a surprising number of terms, and asks for an explanation. - N. J. A. Sloane, May 19 2013

J. McKay, personal communication to N. J. A. Sloane, circa 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cf. A006907, A061256.

A006908 := n -> Sum(Irr(CharacterTable("Symmetric", n)), chi -> Number(chi, x->x<>0)); # Eric M. Schmidt, Jul 13 2012, revised Sep 05 2012

a[n_] := Count[FiniteGroupData[{"SymmetricGroup", n}, "CharacterTable"], k_ /; k != 0, 2]; Array[a, 10] (* Jean-François Alcover, Oct 21 2016 *)

More terms from Eric M. Schmidt, Jul 13 2012

A274691 Number of odd entries in the character table of the symmetric group S_n.

Original entry on oeis.org

1, 1, 4, 7, 19, 33, 77, 135, 218, 392, 798, 1312, 2381, 4107, 6639, 11722, 15869, 26333, 45115, 69168, 106213, 170710, 244042, 384991, 592859, 895944, 1326012, 2055454, 2884762, 4466493, 6553384, 9798596, 13336991, 20192347, 28680574, 41695293, 59766105, 86344867

Offset: 0

Richard Stanley, Jul 02 2016

nonn

			For n = 2, all four character values are 1 or -1, so a(2) = 4.

Hugo Pfoertner, Table of n, a(n) for n = 0..76
Alexander R. Miller, On parity and characters of symmetric groups, preprint.
Alexander R. Miller, On parity and characters of symmetric groups, J. Combin. Theory Ser. A 162 (2019), 231-240. Gives terms for n <= 76.

Cf. A001255, A006907, A335686.

with(combinat):
a:= n-> add(`if`(i[]::odd, 1, 0), i=entries(character(n))):
seq(a(n), n=0..15);  # Alois P. Heinz, Jul 10 2016

More terms from Alois P. Heinz, Jul 10 2016

Further terms from Miller (2019) added by N. J. A. Sloane, Jul 07 2020

A335686 Number of even entries in the character table of the symmetric group S_n.

Original entry on oeis.org

0, 0, 0, 2, 6, 16, 44, 90, 266, 508, 966, 1824, 3548, 6094, 11586, 19254, 37492, 61876, 103110, 170932, 286916, 456554, 759962, 1190034, 1887766, 2937820, 4608084, 7004646, 10938762, 16372732, 24851432, 37014368, 56368810, 82688102, 122855526, 179808396, 263406424

Offset: 0

N. J. A. Sloane, Jul 07 2020

nonn

Hugo Pfoertner, Table of n, a(n) for n = 0..76
Alexander R. Miller, On parity and characters of symmetric groups, preprint.
Alexander R. Miller, On parity and characters of symmetric groups, J. Combin. Theory Ser. A 162 (2019), 231-240. Gives terms for n <= 76.

Cf. A006907, A274691.

A061220 Least entry in character table of the symmetric group S_n.

Original entry on oeis.org

1, -1, -1, -1, -2, -3, -6, -16, -36, -91, -224, -768, -2420, -7854, -22815, -73008, -292864, -1223040, -5002998, -17592960, -67184000, -279734796, -1183614120, -5844883968, -29448258840, -124619677182, -573333075000, -2764864302200, -13664438287500

Offset: 1

Ola Veshta (olaveshta(AT)my-deja.com), May 30 2001

The maximal value in the character table is the maximal degree of an irreducible representation of S_n and this is in sequence A003040.

			a(3) = -1 because the character table of S_3 is / 1 1 1 / 1 1 -1 / 2 -1 0 /.

Eric M. Schmidt, Table of n, a(n) for n = 1..35

Cf. A003040, A006907.

A061220 := n -> Minimum(List(Irr(CharacterTable("Symmetric", n)), Minimum)); # Eric M. Schmidt, Feb 18 2013

seq(min(map(op,[entries(combinat:-character(n))])),n=1..23); # Robert Israel, Mar 31 2016

a[n_] := With[{S = "S" <> ToString[n]}, FiniteGroupData[S, "CharacterTable"] // Flatten // Min]; Array[a, 10] (* Jean-François Alcover, Mar 31 2016 *)

Corrected and extended by Vladeta Jovovic, May 20 2003

More terms from Eric M. Schmidt, Feb 18 2013

cp's OEIS Frontend

A001255 Squares of partition numbers.

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A051748 Number of character table entries of the symmetric group S_n which are < 0.

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A051749 Number of character table entries of the symmetric group S_n which are > 0.

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A006908 Number of nonzero elements in the character table of the symmetric group S_n.

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A274691 Number of odd entries in the character table of the symmetric group S_n.

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A335686 Number of even entries in the character table of the symmetric group S_n.

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A061220 Least entry in character table of the symmetric group S_n.

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