Tiara number triangles

Studies of Boolean functions
sequences related to Boolean functions

A tiara is a number triangle whose entries for row $n$ correspond to the weight of an $n$ -ary Boolean function.
The weight is often shown as an integer $k$ , but it is essentially a fraction ${\frac {k}{n}}$ . E.g. balanced BF have weight ${\frac {1}{2}}$ , and the tautology has weight 1.

The column index can also represent the nonlinearity of a BF. This leads to the unusual tiara Amber, with non-zero enries only in the left half.

An important property of a tiara is the regular triangle of unit fractions, which shall be called its slab. They are shown in the boxes below. See also the example for Emerald.
It is not usually an otherwise known triangle, but for Ruby it is ExPascal, and for Opal it is A289537 with row sums A182176 (number of affine subspaces of F₂ⁿ.)

The names of tiaras and other sequences in this article are likely to be changed again.

The most obvious tiara is that of all Boolean functions. Row $n$ is row $2^{n}$ of Pascal's triangle, and the row sums are $2^{2^{n}}$ . In the triangle of unit fractions $T(n,k)={\binom {2^{n}}{2^{n-k}}}$ .

tiara Clay row sums Grass (A001146)

integer weight

w a	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	sums
0	1	1																																2
1	1	2	1																															4
2	1	4	6	4	1																													16
3	1	8	28	56	70	56	28	8	1																									256
4	1	16	120	560	1820	4368	8008	11440	12870	11440	8008	4368	1820	560	120	16	1																	65536
5	1	32	496	4960	35960	201376	906192	3365856	10518300	28048800	64512240	129024480	225792840	347373600	471435600	565722720	601080390	565722720	471435600	347373600	225792840	129024480	64512240	28048800	10518300	3365856	906192	201376	35960	4960	496	32	1	4294967296

rational weight

w a	$0$	${\frac {1}{32}}$	${\frac {1}{16}}$	${\frac {3}{32}}$	${\frac {1}{8}}$	${\frac {5}{32}}$	${\frac {3}{16}}$	${\frac {7}{32}}$	${\frac {1}{4}}$	${\frac {9}{32}}$	${\frac {5}{16}}$	${\frac {11}{32}}$	${\frac {3}{8}}$	${\frac {13}{32}}$	${\frac {7}{16}}$	${\frac {15}{32}}$	${\frac {1}{2}}$	${\frac {17}{32}}$	${\frac {9}{16}}$	${\frac {19}{32}}$	${\frac {5}{8}}$	${\frac {21}{32}}$	${\frac {11}{16}}$	${\frac {23}{32}}$	${\frac {3}{4}}$	${\frac {25}{32}}$	${\frac {13}{16}}$	${\frac {27}{32}}$	${\frac {7}{8}}$	${\frac {29}{32}}$	${\frac {15}{16}}$	${\frac {31}{32}}$	$1$	sums
0	1																																1	2
1	1																2																1	4
2	1								4								6								4								1	16
3	1				8				28				56				70				56				28				8				1	256
4	1		16		120		560		1820		4368		8008		11440		12870		11440		8008		4368		1820		560		120		16		1	65536
5	1	32	496	4960	35960	201376	906192	3365856	10518300	28048800	64512240	129024480	225792840	347373600	471435600	565722720	601080390	565722720	471435600	347373600	225792840	129024480	64512240	28048800	10518300	3365856	906192	201376	35960	4960	496	32	1	4294967296

unit fractions

w a	$1$	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	${\frac {1}{32}}$	sums
0	1						1
1	1	2					3
2	1	6	4				11
3	1	70	28	8			107
4	1	12870	1820	120	16		14827
5	1	601080390	10518300	35960	496	32	611635179

plain text

rows:
[ 1, 1],
[ 1, 2, 1],
[ 1, 4, 6, 4, 1],
[ 1, 8, 28, 56, 70, 56, 28, 8, 1],
[ 1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1],
[ 1, 32, 496, 4960, 35960, 201376, 906192, 3365856, 10518300, 28048800, 64512240, 129024480, 225792840, 347373600, 471435600, 565722720, 601080390, 565722720, 471435600, 347373600, 225792840, 129024480, 64512240, 28048800, 10518300, 3365856, 906192, 201376, 35960, 4960, 496, 32, 1]

central values: (rational weight = 1/2)
[2, 6, 70, 12870, 601080390]

diagonal: (integer weight = arity)
[ 1, 2, 6, 56, 1820, 201376]

row sums:
[2, 4, 16, 256, 65536, 4294967296]

unit fractions:
[1],
[1, 2],
[1, 6, 4],
[1, 70, 28, 8],
[1, 12870, 1820, 120, 16],
[1, 601080390, 10518300, 35960, 496, 32]

row sums of unit fractions: [1, 3, 11, 107, 14827, 611635179]

An important subset are the dense BF, usually called non-degenerate:

tiara Onyx row sums Lotus (A000371)

integer weight

w a	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	sums
0	1	1																																2
1	0	1	1																															2
2	0	1	4	4	1																													10
3	0	1	13	44	67	56	28	8	1																									218
4	0	1	40	360	1546	4144	7896	11408	12866	11440	8008	4368	1820	560	120	16	1																	64594
5	0	1	121	2680	27550	180096	866432	3308736	10453960	27991600	64472200	129002640	225783740	347370800	471435000	565722640	601080385	565722720	471435600	347373600	225792840	129024480	64512240	28048800	10518300	3365856	906192	201376	35960	4960	496	32	1	4294642034

rational weight

w a	$0$	${\frac {1}{32}}$	${\frac {1}{16}}$	${\frac {3}{32}}$	${\frac {1}{8}}$	${\frac {5}{32}}$	${\frac {3}{16}}$	${\frac {7}{32}}$	${\frac {1}{4}}$	${\frac {9}{32}}$	${\frac {5}{16}}$	${\frac {11}{32}}$	${\frac {3}{8}}$	${\frac {13}{32}}$	${\frac {7}{16}}$	${\frac {15}{32}}$	${\frac {1}{2}}$	${\frac {17}{32}}$	${\frac {9}{16}}$	${\frac {19}{32}}$	${\frac {5}{8}}$	${\frac {21}{32}}$	${\frac {11}{16}}$	${\frac {23}{32}}$	${\frac {3}{4}}$	${\frac {25}{32}}$	${\frac {13}{16}}$	${\frac {27}{32}}$	${\frac {7}{8}}$	${\frac {29}{32}}$	${\frac {15}{16}}$	${\frac {31}{32}}$	$1$	sums
0	1																																1	2
1	0																1																1	2
2	0								1								4								4								1	10
3	0				1				13				44				67				56				28				8				1	218
4	0		1		40		360		1546		4144		7896		11408		12866		11440		8008		4368		1820		560		120		16		1	64594
5	0	1	121	2680	27550	180096	866432	3308736	10453960	27991600	64472200	129002640	225783740	347370800	471435000	565722640	601080385	565722720	471435600	347373600	225792840	129024480	64512240	28048800	10518300	3365856	906192	201376	35960	4960	496	32	1	4294642034

unit fractions

w a	$1$	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	${\frac {1}{32}}$	sums
0	1						1
1	1	1					2
2	1	4	1				6
3	1	67	13	1			82
4	1	12866	1546	40	1		14454
5	1	601080385	10453960	27550	121	1	611562018

plain text

rows:
[ 1, 1],
[ 0, 1, 1],
[ 0, 1, 4, 4, 1],
[ 0, 1, 13, 44, 67, 56, 28, 8, 1],
[ 0, 1, 40, 360, 1546, 4144, 7896, 11408, 12866, 11440, 8008, 4368, 1820, 560, 120, 16, 1],
[ 0, 1, 121, 2680, 27550, 180096, 866432, 3308736, 10453960, 27991600, 64472200, 129002640, 225783740, 347370800, 471435000, 565722640, 601080385, 565722720, 471435600, 347373600, 225792840, 129024480, 64512240, 28048800, 10518300, 3365856, 906192, 201376, 35960, 4960, 496, 32, 1]

central values: (rational weight = 1/2)
[1, 4, 67, 12866, 601080385]

diagonal: (integer weight = arity)
[ 1, 1, 4, 44, 1546, 180096]

row sums:
[2, 2, 10, 218, 64594, 4294642034]

unit fractions:
[1],
[1, 1],
[1, 4, 1],
[1, 67, 13, 1],
[1, 12866, 1546, 40, 1],
[1, 601080385, 10453960, 27550, 121, 1]

row sums of unit fractions: [1, 2, 6, 82, 14454, 611562018]

equivalence classes

numbers of families

The central column in Saffron.

tiara Emerald (A054724) row sums Thyme (A000231)

integer weight

w a	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	sums
0	1	1																																2
1	1	1	1																															3
2	1	1	3	1	1																													7
3	1	1	7	7	14	7	7	1	1																									46
4	1	1	15	35	140	273	553	715	870	715	553	273	140	35	15	1	1																	4336
5	1	1	31	155	1240	6293	28861	105183	330460	876525	2020239	4032015	7063784	10855425	14743445	17678835	18796230	17678835	14743445	10855425	7063784	4032015	2020239	876525	330460	105183	28861	6293	1240	155	31	1	1	134281216

rational weight

w a	$0$	${\frac {1}{32}}$	${\frac {1}{16}}$	${\frac {3}{32}}$	${\frac {1}{8}}$	${\frac {5}{32}}$	${\frac {3}{16}}$	${\frac {7}{32}}$	${\frac {1}{4}}$	${\frac {9}{32}}$	${\frac {5}{16}}$	${\frac {11}{32}}$	${\frac {3}{8}}$	${\frac {13}{32}}$	${\frac {7}{16}}$	${\frac {15}{32}}$	${\frac {1}{2}}$	${\frac {17}{32}}$	${\frac {9}{16}}$	${\frac {19}{32}}$	${\frac {5}{8}}$	${\frac {21}{32}}$	${\frac {11}{16}}$	${\frac {23}{32}}$	${\frac {3}{4}}$	${\frac {25}{32}}$	${\frac {13}{16}}$	${\frac {27}{32}}$	${\frac {7}{8}}$	${\frac {29}{32}}$	${\frac {15}{16}}$	${\frac {31}{32}}$	$1$	sums
0	1																																1	2
1	1																1																1	3
2	1								1								3								1								1	7
3	1				1				7				7				14				7				7				1				1	46
4	1		1		15		35		140		273		553		715		870		715		553		273		140		35		15		1		1	4336
5	1	1	31	155	1240	6293	28861	105183	330460	876525	2020239	4032015	7063784	10855425	14743445	17678835	18796230	17678835	14743445	10855425	7063784	4032015	2020239	876525	330460	105183	28861	6293	1240	155	31	1	1	134281216

unit fractions

w a	$1$	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	${\frac {1}{32}}$	sums
0	1						1
1	1	1					2
2	1	3	1				5
3	1	14	7	1			23
4	1	870	140	15	1		1027
5	1	18796230	330460	1240	31	1	19127963

plain text

rows:
[ 1, 1],
[ 1, 1, 1],
[ 1, 1, 3, 1, 1],
[ 1, 1, 7, 7, 14, 7, 7, 1, 1],
[ 1, 1, 15, 35, 140, 273, 553, 715, 870, 715, 553, 273, 140, 35, 15, 1, 1],
[ 1, 1, 31, 155, 1240, 6293, 28861, 105183, 330460, 876525, 2020239, 4032015, 7063784, 10855425, 14743445, 17678835, 18796230, 17678835, 14743445, 10855425, 7063784, 4032015, 2020239, 876525, 330460, 105183, 28861, 6293, 1240, 155, 31, 1, 1]

central values: (rational weight = 1/2)
[1, 3, 14, 870, 18796230]

diagonal: (integer weight = arity)
[ 1, 1, 3, 7, 140, 6293]

row sums:
[2, 3, 7, 46, 4336, 134281216]

unit fractions:
[1],
[1, 1],
[1, 3, 1],
[1, 14, 7, 1],
[1, 870, 140, 15, 1],
[1, 18796230, 330460, 1240, 31, 1]

row sums of unit fractions: [1, 2, 5, 23, 1027, 19127963]

numbers of diploid families

$T(n,k)$ is the number of $k$ -subsets of $\{0...2^{n}-1\}$ whose bitwise XOR is 0. The row sums are $2^{2^{n}-n}$ .
E.g. T(4, 3) = 35 is the number of 3-subsets of 0...15 whose bitwise XOR is 0.
Those are: (1, 2, 3), (1, 4, 5), (1, 6, 7), (1, 8, 9), (1, 10, 11), (1, 12, 13), (1, 14, 15), (2, 4, 6), (2, 5, 7), (2, 8, 10), (2, 9, 11), (2, 12, 14), (2, 13, 15), (3, 4, 7), (3, 5, 6), (3, 8, 11), (3, 9, 10), (3, 12, 15), (3, 13, 14), (4, 8, 12), (4, 9, 13), (4, 10, 14), (4, 11, 15), (5, 8, 13), (5, 9, 12), (5, 10, 15), (5, 11, 14), (6, 8, 14), (6, 9, 15), (6, 10, 12), (6, 11, 13), (7, 8, 15), (7, 9, 14), (7, 10, 13), (7, 11, 12)
The central column is A340259.

tiara Amethyst (A340312) row sums A300361

integer weight

w a	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	sums
0	1	1																																2
1	1	1	0																															2
2	1	1	0	1	1																													4
3	1	1	0	7	14	7	0	1	1																									32
4	1	1	0	35	140	273	448	715	870	715	448	273	140	35	0	1	1																	4096
5	1	1	0	155	1240	6293	27776	105183	330460	876525	2011776	4032015	7063784	10855425	14721280	17678835	18796230	17678835	14721280	10855425	7063784	4032015	2011776	876525	330460	105183	27776	6293	1240	155	0	1	1	134217728

rational weight

w a	$0$	${\frac {1}{32}}$	${\frac {1}{16}}$	${\frac {3}{32}}$	${\frac {1}{8}}$	${\frac {5}{32}}$	${\frac {3}{16}}$	${\frac {7}{32}}$	${\frac {1}{4}}$	${\frac {9}{32}}$	${\frac {5}{16}}$	${\frac {11}{32}}$	${\frac {3}{8}}$	${\frac {13}{32}}$	${\frac {7}{16}}$	${\frac {15}{32}}$	${\frac {1}{2}}$	${\frac {17}{32}}$	${\frac {9}{16}}$	${\frac {19}{32}}$	${\frac {5}{8}}$	${\frac {21}{32}}$	${\frac {11}{16}}$	${\frac {23}{32}}$	${\frac {3}{4}}$	${\frac {25}{32}}$	${\frac {13}{16}}$	${\frac {27}{32}}$	${\frac {7}{8}}$	${\frac {29}{32}}$	${\frac {15}{16}}$	${\frac {31}{32}}$	$1$	sums
0	1																																1	2
1	1																1																0	2
2	1								1								0								1								1	4
3	1				1				0				7				14				7				0				1				1	32
4	1		1		0		35		140		273		448		715		870		715		448		273		140		35		0		1		1	4096
5	1	1	0	155	1240	6293	27776	105183	330460	876525	2011776	4032015	7063784	10855425	14721280	17678835	18796230	17678835	14721280	10855425	7063784	4032015	2011776	876525	330460	105183	27776	6293	1240	155	0	1	1	134217728

unit fractions

w a	$1$	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	${\frac {1}{32}}$	sums
0	1						1
1	0	1					1
2	1	0	1				2
3	1	14	0	1			16
4	1	870	140	0	1		1012
5	1	18796230	330460	1240	0	1	19127932

plain text

rows:
[ 1, 1],
[ 1, 1, 0],
[ 1, 1, 0, 1, 1],
[ 1, 1, 0, 7, 14, 7, 0, 1, 1],
[ 1, 1, 0, 35, 140, 273, 448, 715, 870, 715, 448, 273, 140, 35, 0, 1, 1],
[ 1, 1, 0, 155, 1240, 6293, 27776, 105183, 330460, 876525, 2011776, 4032015, 7063784, 10855425, 14721280, 17678835, 18796230, 17678835, 14721280, 10855425, 7063784, 4032015, 2011776, 876525, 330460, 105183, 27776, 6293, 1240, 155, 0, 1, 1]

central values: (rational weight = 1/2)
[1, 0, 14, 870, 18796230]

diagonal: (integer weight = arity)
[ 1, 1, 0, 7, 140, 6293]

row sums:
[2, 2, 4, 32, 4096, 134217728]

unit fractions:
[1],
[0, 1],
[1, 0, 1],
[1, 14, 0, 1],
[1, 870, 140, 0, 1],
[1, 18796230, 330460, 1240, 0, 1]

row sums of unit fractions: [1, 1, 2, 16, 1012, 19127932]

numbers of clans

The central column is A000721.

tiara Diamond (A039754) row sums A000616

integer weight

w a	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	sums
0	1	1																																2
1	1	1	1																															3
2	1	1	2	1	1																													6
3	1	1	3	3	6	3	3	1	1																									22
4	1	1	4	6	19	27	50	56	74	56	50	27	19	6	4	1	1																	402
5	1	1	5	10	47	131	472	1326	3779	9013	19963	38073	65664	98804	133576	158658	169112	158658	133576	98804	65664	38073	19963	9013	3779	1326	472	131	47	10	5	1	1	1228158

rational weight

w a	$0$	${\frac {1}{32}}$	${\frac {1}{16}}$	${\frac {3}{32}}$	${\frac {1}{8}}$	${\frac {5}{32}}$	${\frac {3}{16}}$	${\frac {7}{32}}$	${\frac {1}{4}}$	${\frac {9}{32}}$	${\frac {5}{16}}$	${\frac {11}{32}}$	${\frac {3}{8}}$	${\frac {13}{32}}$	${\frac {7}{16}}$	${\frac {15}{32}}$	${\frac {1}{2}}$	${\frac {17}{32}}$	${\frac {9}{16}}$	${\frac {19}{32}}$	${\frac {5}{8}}$	${\frac {21}{32}}$	${\frac {11}{16}}$	${\frac {23}{32}}$	${\frac {3}{4}}$	${\frac {25}{32}}$	${\frac {13}{16}}$	${\frac {27}{32}}$	${\frac {7}{8}}$	${\frac {29}{32}}$	${\frac {15}{16}}$	${\frac {31}{32}}$	$1$	sums
0	1																																1	2
1	1																1																1	3
2	1								1								2								1								1	6
3	1				1				3				3				6				3				3				1				1	22
4	1		1		4		6		19		27		50		56		74		56		50		27		19		6		4		1		1	402
5	1	1	5	10	47	131	472	1326	3779	9013	19963	38073	65664	98804	133576	158658	169112	158658	133576	98804	65664	38073	19963	9013	3779	1326	472	131	47	10	5	1	1	1228158

unit fractions

w a	$1$	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	${\frac {1}{32}}$	sums
0	1						1
1	1	1					2
2	1	2	1				4
3	1	6	3	1			11
4	1	74	19	4	1		99
5	1	169112	3779	47	5	1	172945

plain text

rows:
[ 1, 1],
[ 1, 1, 1],
[ 1, 1, 2, 1, 1],
[ 1, 1, 3, 3, 6, 3, 3, 1, 1],
[ 1, 1, 4, 6, 19, 27, 50, 56, 74, 56, 50, 27, 19, 6, 4, 1, 1],
[ 1, 1, 5, 10, 47, 131, 472, 1326, 3779, 9013, 19963, 38073, 65664, 98804, 133576, 158658, 169112, 158658, 133576, 98804, 65664, 38073, 19963, 9013, 3779, 1326, 472, 131, 47, 10, 5, 1, 1]

central values: (rational weight = 1/2)
[1, 2, 6, 74, 169112]

diagonal: (integer weight = arity)
[ 1, 1, 2, 3, 19, 131]

row sums:
[2, 3, 6, 22, 402, 1228158]

unit fractions:
[1],
[1, 1],
[1, 2, 1],
[1, 6, 3, 1],
[1, 74, 19, 4, 1],
[1, 169112, 3779, 47, 5, 1]

row sums of unit fractions: [1, 2, 4, 11, 99, 172945]

numbers of factions

tiara Garnet (A052265) row sums DubAnise (A003180)

integer weight

w a	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	sums
0	1	1																																2
1	1	2	1																															4
2	1	3	4	3	1																													12
3	1	4	9	16	20	16	9	4	1																									80
4	1	5	17	52	136	284	477	655	730	655	477	284	136	52	17	5	1																	3984
5	1	6	28	134	625	2674	10195	34230	100577	258092	579208	1140090	1974438	3016994	4077077	4881092	5182326	4881092	4077077	3016994	1974438	1140090	579208	258092	100577	34230	10195	2674	625	134	28	6	1	37333248

rational weight

w a	$0$	${\frac {1}{32}}$	${\frac {1}{16}}$	${\frac {3}{32}}$	${\frac {1}{8}}$	${\frac {5}{32}}$	${\frac {3}{16}}$	${\frac {7}{32}}$	${\frac {1}{4}}$	${\frac {9}{32}}$	${\frac {5}{16}}$	${\frac {11}{32}}$	${\frac {3}{8}}$	${\frac {13}{32}}$	${\frac {7}{16}}$	${\frac {15}{32}}$	${\frac {1}{2}}$	${\frac {17}{32}}$	${\frac {9}{16}}$	${\frac {19}{32}}$	${\frac {5}{8}}$	${\frac {21}{32}}$	${\frac {11}{16}}$	${\frac {23}{32}}$	${\frac {3}{4}}$	${\frac {25}{32}}$	${\frac {13}{16}}$	${\frac {27}{32}}$	${\frac {7}{8}}$	${\frac {29}{32}}$	${\frac {15}{16}}$	${\frac {31}{32}}$	$1$	sums
0	1																																1	2
1	1																2																1	4
2	1								3								4								3								1	12
3	1				4				9				16				20				16				9				4				1	80
4	1		5		17		52		136		284		477		655		730		655		477		284		136		52		17		5		1	3984
5	1	6	28	134	625	2674	10195	34230	100577	258092	579208	1140090	1974438	3016994	4077077	4881092	5182326	4881092	4077077	3016994	1974438	1140090	579208	258092	100577	34230	10195	2674	625	134	28	6	1	37333248

unit fractions

w a	$1$	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	${\frac {1}{32}}$	sums
0	1						1
1	1	2					3
2	1	4	3				8
3	1	20	9	4			34
4	1	730	136	17	5		889
5	1	5182326	100577	625	28	6	5283563

plain text

rows:
[ 1, 1],
[1, 2, 1],
[1, 3, 4, 3, 1],
[1, 4, 9, 16, 20, 16, 9, 4, 1],
[1, 5, 17, 52, 136, 284, 477, 655, 730, 655, 477, 284, 136, 52, 17, 5, 1],
[1, 6, 28, 134, 625, 2674, 10195, 34230, 100577, 258092, 579208, 1140090, 1974438, 3016994, 4077077, 4881092, 5182326, 4881092, 4077077, 3016994, 1974438, 1140090, 579208, 258092, 100577, 34230, 10195, 2674, 625, 134, 28, 6, 1 ]

central values: (rational weight = 1/2)
[2, 4, 20, 730, 5182326]

diagonal: (integer weight = arity)
[ 1, 2, 4, 16, 136, 2674]

row sums:
[2, 4, 12, 80, 3984, 37333248]

unit fractions:
[1],
[1, 2],
[1, 4, 3],
[1, 20, 9, 4],
[1, 730, 136, 17, 5],
[1 , 5182326, 100577, 625, 28, 6]

row sums of unit fractions: [1, 3, 8, 34, 889, 5283563]

oddacity and gender

Gender of Boolean functions § oddacity and gender

	0	1	2	3	4
female	1	1	3	97	32199
evenacious female	1	1	1	57	30537
oddacious female	0	0	2	40	1662
oddacious male	1	3	13	159	33337
oddacious	1	3	15	199	34999
all	2	4	16	256	65536

tiara NonRuby row sums Primula (A246537)

integer weight

w a	0	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	sums
0	1																1
1	1	0															1
2	1	2	0	0													3
3	1	16	0	64	0	16	0	0									97
4	1	88	0	1796	0	7784	0	12862	0	7784	0	1796	0	88	0	0	32199

rational weight

w a	$0$	${\frac {1}{16}}$	${\frac {1}{8}}$	${\frac {3}{16}}$	${\frac {1}{4}}$	${\frac {5}{16}}$	${\frac {3}{8}}$	${\frac {7}{16}}$	${\frac {1}{2}}$	${\frac {9}{16}}$	${\frac {5}{8}}$	${\frac {11}{16}}$	${\frac {3}{4}}$	${\frac {13}{16}}$	${\frac {7}{8}}$	${\frac {15}{16}}$	sums
0	1																1
1	1								0								1
2	1				0				2				0				3
3	1		0		16		0		64		0		16		0		97
4	1	0	88	0	1796	0	7784	0	12862	0	7784	0	1796	0	88	0	32199

unit fractions

w a	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	sums
0					0
1	0				0
2	2	0			2
3	64	16	0		80
4	12862	1796	88	0	14746

plain text

rows:
[ 1, 0],
[1, 0, 0],
[1, 0, 2, 0, 0],
[1, 0, 16, 0, 64, 0, 16, 0, 0],
[1, 0, 88, 0, 1796, 0, 7784, 0, 12862, 0, 7784, 0, 1796, 0, 88, 0, 0 ]

central values: (rational weight = 1/2)
[0, 2, 64, 12862]

diagonal: (integer weight = arity)
[ 1, 0, 2, 0, 1796]

row sums:
[1, 1, 3, 97, 32199]

unit fractions:
[0],
[0, 0],
[0, 2, 0],
[0, 64, 16, 0],
[0 , 12862, 1796, 88, 0]

row sums of unit fractions: [0, 0, 2, 80, 14746]

tiara NonOpal

integer weight

w a	0	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	sums
0	1																1
1	1	0															1
2	1	0	0	0													1
3	1	0	0	56	0	0	0	0									57
4	1	0	0	1680	0	7168	0	12840	0	7168	0	1680	0	0	0	0	30537

rational weight

w a	$0$	${\frac {1}{16}}$	${\frac {1}{8}}$	${\frac {3}{16}}$	${\frac {1}{4}}$	${\frac {5}{16}}$	${\frac {3}{8}}$	${\frac {7}{16}}$	${\frac {1}{2}}$	${\frac {9}{16}}$	${\frac {5}{8}}$	${\frac {11}{16}}$	${\frac {3}{4}}$	${\frac {13}{16}}$	${\frac {7}{8}}$	${\frac {15}{16}}$	sums
0	1																1
1	1								0								1
2	1				0				0				0				1
3	1		0		0		0		56		0		0		0		57
4	1	0	0	0	1680	0	7168	0	12840	0	7168	0	1680	0	0	0	30537

unit fractions

w a	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	sums
0					0
1	0				0
2	0	0			0
3	56	0	0		56
4	12840	1680	0	0	14520

plain text

rows:
[ 1, 0],
[1, 0, 0],
[1, 0, 0, 0, 0],
[1, 0, 0, 0, 56, 0, 0, 0, 0],
[1, 0, 0, 0, 1680, 0, 7168, 0, 12840, 0, 7168, 0, 1680, 0, 0, 0, 0 ]

central values: (rational weight = 1/2)
[0, 0, 56, 12840]

diagonal: (integer weight = arity)
[ 1, 0, 0, 0, 1680]

row sums:
[1, 1, 1, 57, 30537]

unit fractions:
[0],
[0, 0],
[0, 0, 0],
[0, 56, 0, 0],
[0 , 12840, 1680, 0, 0]

row sums of unit fractions: [0, 0, 0, 56, 14520]

tiara Jade

integer weight

w a	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	sums
0																0
1	0															0
2	2	0	0													2
3	16	0	8	0	16	0	0									40
4	88	0	116	0	616	0	22	0	616	0	116	0	88	0	0	1662

rational weight

w a	${\frac {1}{16}}$	${\frac {1}{8}}$	${\frac {3}{16}}$	${\frac {1}{4}}$	${\frac {5}{16}}$	${\frac {3}{8}}$	${\frac {7}{16}}$	${\frac {1}{2}}$	${\frac {9}{16}}$	${\frac {5}{8}}$	${\frac {11}{16}}$	${\frac {3}{4}}$	${\frac {13}{16}}$	${\frac {7}{8}}$	${\frac {15}{16}}$	sums
0																0
1								0								0
2				0				2				0				2
3		0		16		0		8		0		16		0		40
4	0	88	0	116	0	616	0	22	0	616	0	116	0	88	0	1662

unit fractions

w a	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	sums
0					0
1	0				0
2	2	0			2
3	8	16	0		24
4	22	116	88	0	226

plain text

rows:
[ 0, 0],
[0, 0, 0],
[0, 0, 2, 0, 0],
[0, 0, 16, 0, 8, 0, 16, 0, 0],
[0, 0, 88, 0, 116, 0, 616, 0, 22, 0, 616, 0, 116, 0, 88, 0, 0 ]

central values: (rational weight = 1/2)
[0, 2, 8, 22]

diagonal: (integer weight = arity)
[ 0, 0, 2, 0, 116]

row sums:
[0, 0, 2, 40, 1662]

unit fractions:
[0],
[0, 0],
[0, 2, 0],
[0, 8, 16, 0],
[0 , 22, 116, 88, 0]

row sums of unit fractions: [0, 0, 2, 24, 226]

tiara Ruby row sums HalfCrocus (A246418)

integer weight

w a	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	sums
0	1																1
1	2	1															3
2	4	4	4	1													13
3	8	12	56	6	56	12	8	1									159
4	16	32	560	24	4368	224	11440	8	11440	224	4368	24	560	32	16	1	33337

rational weight

w a	${\frac {1}{16}}$	${\frac {1}{8}}$	${\frac {3}{16}}$	${\frac {1}{4}}$	${\frac {5}{16}}$	${\frac {3}{8}}$	${\frac {7}{16}}$	${\frac {1}{2}}$	${\frac {9}{16}}$	${\frac {5}{8}}$	${\frac {11}{16}}$	${\frac {3}{4}}$	${\frac {13}{16}}$	${\frac {7}{8}}$	${\frac {15}{16}}$	$1$	sums
0																1	1
1								2								1	3
2				4				4				4				1	13
3		8		12		56		6		56		12		8		1	159
4	16	32	560	24	4368	224	11440	8	11440	224	4368	24	560	32	16	1	33337

unit fractions

w a	$1$	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	sums
0	1					1
1	1	2				3
2	1	4	4			9
3	1	6	12	8		27
4	1	8	24	32	16	81

plain text

rows:
[ 0, 1],
[0, 2, 1],
[0, 4, 4, 4, 1],
[0, 8, 12, 56, 6, 56, 12, 8, 1],
[0, 16, 32, 560, 24, 4368, 224, 11440, 8, 11440, 224, 4368, 24, 560, 32, 16, 1 ]

central values: (rational weight = 1/2)
[2, 4, 6, 8]

diagonal: (integer weight = arity)
[ 0, 2, 4, 56, 24]

row sums:
[1, 3, 13, 159, 33337]

unit fractions:
[1],
[1, 2],
[1, 4, 4],
[1, 6, 12, 8],
[1 , 8, 24, 32, 16]

row sums of unit fractions: [1, 3, 9, 27, 81]

tiara Opal

integer weight

w a	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	sums
0	1																1
1	2	1															3
2	4	6	4	1													15
3	8	28	56	14	56	28	8	1									199
4	16	120	560	140	4368	840	11440	30	11440	840	4368	140	560	120	16	1	34999

rational weight

w a	${\frac {1}{16}}$	${\frac {1}{8}}$	${\frac {3}{16}}$	${\frac {1}{4}}$	${\frac {5}{16}}$	${\frac {3}{8}}$	${\frac {7}{16}}$	${\frac {1}{2}}$	${\frac {9}{16}}$	${\frac {5}{8}}$	${\frac {11}{16}}$	${\frac {3}{4}}$	${\frac {13}{16}}$	${\frac {7}{8}}$	${\frac {15}{16}}$	$1$	sums
0																1	1
1								2								1	3
2				4				6				4				1	15
3		8		28		56		14		56		28		8		1	199
4	16	120	560	140	4368	840	11440	30	11440	840	4368	140	560	120	16	1	34999

unit fractions

w a	$1$	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	sums
0	1					1
1	1	2				3
2	1	6	4			11
3	1	14	28	8		51
4	1	30	140	120	16	307

plain text

rows:
[ 0, 1],
[0, 2, 1],
[0, 4, 6, 4, 1],
[0, 8, 28, 56, 14, 56, 28, 8, 1],
[0, 16, 120, 560, 140, 4368, 840, 11440, 30, 11440, 840, 4368, 140, 560, 120, 16, 1 ]

central values: (rational weight = 1/2)
[2, 6, 14, 30]

diagonal: (integer weight = arity)
[ 0, 2, 6, 56, 140]

row sums:
[1, 3, 15, 199, 34999]

unit fractions:
[1],
[1, 2],
[1, 6, 4],
[1, 14, 28, 8],
[1 , 30, 140, 120, 16]

row sums of unit fractions: [1, 3, 11, 51, 307]

mentor permutation

These tiaras count the fixed points of the mentor permutation. (Both have the same row sums.)
Flint is for the permutation between Zhegalkin indices, and Slate for the one between truth tables. The latter is symmetric.

tiara Flint

integer weight

w a	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	sums
0	1	1																2
1	1	1	0															2
2	1	4	6	4	1													16
3	1	4	9	16	19	12	3	0	0									64
4	1	5	25	85	170	226	226	170	85	25	5	1	0	0	0	0	0	1024

rational weight

w a	$0$	${\frac {1}{16}}$	${\frac {1}{8}}$	${\frac {3}{16}}$	${\frac {1}{4}}$	${\frac {5}{16}}$	${\frac {3}{8}}$	${\frac {7}{16}}$	${\frac {1}{2}}$	${\frac {9}{16}}$	${\frac {5}{8}}$	${\frac {11}{16}}$	${\frac {3}{4}}$	${\frac {13}{16}}$	${\frac {7}{8}}$	${\frac {15}{16}}$	$1$	sums
0	1																1	2
1	1								1								0	2
2	1				4				6				4				1	16
3	1		4		9		16		19		12		3		0		0	64
4	1	5	25	85	170	226	226	170	85	25	5	1	0	0	0	0	0	1024

unit fractions

w a	$1$	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	sums
0	1					1
1	0	1				1
2	1	6	4			11
3	0	19	9	4		32
4	0	85	170	25	5	285

plain text

rows:
[ 1, 1],
[ 1, 1, 0],
[ 1, 4, 6, 4, 1],
[ 1, 4, 9, 16, 19, 12, 3, 0, 0],
[ 1, 5, 25, 85, 170, 226, 226, 170, 85, 25, 5, 1, 0, 0, 0, 0, 0]

central values: (rational weight = 1/2)
[1, 6, 19, 85]

diagonal: (integer weight = arity)
[ 1, 1, 6, 16, 170]

row sums:
[2, 2, 16, 64, 1024]

unit fractions:
[1],
[0, 1],
[1, 6, 4],
[0, 19, 9, 4],
[0, 85, 170, 25, 5]

row sums of unit fractions: [1, 1, 11, 32, 285]

tiara Slate

integer weight

w a	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	sums
0	1	1																2
1	1	0	1															2
2	1	4	6	4	1													16
3	1	0	12	0	38	0	12	0	1									64
4	1	0	0	0	60	0	256	0	390	0	256	0	60	0	0	0	1	1024

rational weight

w a	$0$	${\frac {1}{16}}$	${\frac {1}{8}}$	${\frac {3}{16}}$	${\frac {1}{4}}$	${\frac {5}{16}}$	${\frac {3}{8}}$	${\frac {7}{16}}$	${\frac {1}{2}}$	${\frac {9}{16}}$	${\frac {5}{8}}$	${\frac {11}{16}}$	${\frac {3}{4}}$	${\frac {13}{16}}$	${\frac {7}{8}}$	${\frac {15}{16}}$	$1$	sums
0	1																1	2
1	1								0								1	2
2	1				4				6				4				1	16
3	1		0		12		0		38		0		12		0		1	64
4	1	0	0	0	60	0	256	0	390	0	256	0	60	0	0	0	1	1024

unit fractions

w a	$1$	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	sums
0	1					1
1	1	0				1
2	1	6	4			11
3	1	38	12	0		51
4	1	390	60	0	0	451

plain text

rows:
[ 1, 1],
[ 1, 0, 1],
[ 1, 4, 6, 4, 1],
[ 1, 0, 12, 0, 38, 0, 12, 0, 1],
[ 1, 0, 0, 0, 60, 0, 256, 0, 390, 0, 256, 0, 60, 0, 0, 0, 1]

central values: (rational weight = 1/2)
[0, 6, 38, 390]

diagonal: (integer weight = arity)
[ 1, 0, 6, 0, 60]

row sums:
[2, 2, 16, 64, 1024]

unit fractions:
[1],
[1, 0],
[1, 6, 4],
[1, 38, 12, 0],
[1, 390, 60, 0, 0]

row sums of unit fractions: [1, 1, 11, 51, 451]