User:Ludovic Schwob

I am a PhD student at LIGM, Université Gustave Eiffel, France. I am interested in combinatorics, arithmetics, geometry...

Contributed sequences

Polygons

Created

A309318: Number of polygons whose vertices are the (2*n+1)-th roots of unity and whose 2*n+1 sides all have distinct slopes.

A330660: Number of polygons formed by connecting the vertices of a regular {2*n+1}-gon such that they make k turns around the center point.

A330662: Number of polygons with 2*n sides, of which k run through the center of a circle, on the circumference of which the 2*n vertices of the polygon are arranged at equal spacing.

A342968: Number of n+2-sided polygons with the property that one makes k turns on itself while following its edges.

A343257: Number of n+2-sided polygons whose points lie on a circle and with the property that one makes k turns on itself, always in the same direction (right or left) while following its edges.

A343369: Number of polygons formed by connecting the vertices of a regular 2n-gon such that the winding number around the center is k and with no side passing through the center.

A358328: Number of polygons with 2*n sides, of which k run through the center of a circle, on the circumference of which the 2*n vertices of the polygon are arranged at equal spacing, up to rotation.

A358329: Number of polygons with 2*n sides, of which k run through the center of a circle, on the circumference of which the 2*n vertices of the polygon are arranged at equal spacing, up to rotation and reflection.

A365094: Number of n-sided cycles with the property that one makes k turns to the right while following its edges.

Computed more terms

A295264: Number of total cyclic orders Z on {0, ..., n-1} such that (i, (i+1) mod n, (i+2) mod n) in Z for 0 <= i < n.

Matrices and Young Tableaux

Created

A365961: Number of (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums.

A359856: Number of permutations of [1..n] which are indecomposable by direct and skew sums.

A363689: Number of alternating sign matrices of size n which are indecomposable by direct and skew sums.

Computed more terms

A068313: Number of (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

A178718: Sum of entries of n-th Kostka matrix for the partitions of n.

A224879: Number of equivalence classes of n X n nonsingular matrices over GF(2), up to row and column permutation.

A299968: Number of normal generalized Young tableaux of size n with all rows and columns strictly increasing.

A321652: Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

A321737: Number of ways to partition the Young diagram of an integer partition of n into vertical sections.

A321734: Number of nonnegative integer square matrices with sum of entries equal to n, no zero rows or columns, weakly decreasing row and column sums, and the same row sums as column sums.

A231498: Dimensions of algebraic generators of Hopf algebra ASM.

A231499: Dimensions of totally primitive elements of Hopf algebra ASM.

In the following table, $\lambda \vdash n$ is an integer partition of $n$, $K_{\lambda ,\mu }$ are Kostka numbers and $a_{\lambda }={\frac {(\sum _{i}m_{i})!}{\prod _{i}m_{i}!}}$ where $m_{i}$ are the multiplicities of $\lambda$.

	Number of $\{0,1\}$-Matrices with no zero row and column and sum $n$.	Number of $\mathbf {N}$-Matrices with no zero row and column and sum $n$.
All	1, 4, 24, 196, 2016, 24976, 361792... (A101370) $a(n)=\sum _{\lambda ,\mu ,\nu \vdash n}a_{\mu }a_{\nu }K_{\lambda \mu }K_{{\overline {\lambda }}\nu }$	1, 5, 33, 281, 2961, 37277, 546193... (A120733) $a(n)=\sum _{\lambda ,\mu ,\nu \vdash n}a_{\mu }a_{\nu }K_{\lambda \mu }K_{\lambda \nu }$
Up to permutation of rows	1, 3, 10, 41, 192, 1025, 6087, 39754... (A116540)
Up to permutation of rows and columns	1, 3, 6, 16, 34, 90, 211, 558, 1430, 3908... (A049311)	1, 4, 10, 33, 91, 298, 910, 3017, 9945... (A007716)
Symmetric	1, 2, 6, 20, 74, 302, 1314, 6122, 29982... (A135588)	1, 3, 9, 33, 125, 531, 2349, 11205, 55589... (A138178) $a(n)=\sum _{\lambda ,\mu \vdash n}a_{\mu }K_{\lambda \mu }$
Square	1, 2, 10, 70, 642, 7246, 97052, 1503700... (A104602) $a(n)=\sum _{\underset {l(\mu )=l(\nu )}{\lambda ,\mu ,\nu \vdash n}}a_{\mu }a_{\nu }K_{\lambda \mu }K_{{\overline {\lambda }}\nu }$	1, 3, 15, 107, 991, 11267, 151721... (A120732) $a(n)=\sum _{\underset {l(\mu )=l(\nu )}{\lambda ,\mu ,\nu \vdash n}}a_{\mu }a_{\nu }K_{\lambda \mu }K_{\lambda \nu }$
Equal row and column sums	1, 2, 8, 40, 246, 1816, 15630, 153592... (A321733) $a(n)=\sum _{\lambda ,\mu \vdash n}a_{\mu }K_{\lambda \mu }K_{{\overline {\lambda }}\mu }$	1, 3, 11, 53, 317, 2293, 19435, 188851... (A321732) $a(n)=\sum _{\lambda ,\mu \vdash n}a_{\mu }K_{\lambda \mu }^{2}$
Nonincreasing row sums	1, 4, 19, 127, 967, 9063, 94595, 1139708... (A365961) $a(n)=\sum _{\lambda ,\mu ,\nu \vdash n}a_{\mu }K_{\lambda \mu }K_{{\overline {\lambda }}\nu }$	1, 5, 25, 173, 1297, 12225, 124997... (A035341) $a(n)=\sum _{\lambda ,\mu ,\nu \vdash n}a_{\mu }K_{\lambda \mu }K_{\lambda \nu }$
Nonincreasing row and column sums	1, 4, 15, 82, 457, 3231, 24055, 209375... (A068313) $a(n)=\sum _{\lambda ,\mu ,\nu \vdash n}K_{\lambda \mu }K_{{\overline {\lambda }}\nu }$	1, 5, 19, 107, 573, 4050, 29093, 249301... (A321652) $a(n)=\sum _{\lambda ,\mu ,\nu \vdash n}K_{\lambda \mu }K_{\lambda \nu }$
Nonincreasing and equal row and column sums	1, 2, 7, 30, 153, 939, 6653, 53743... (A321735) $a(n)=\sum _{\lambda ,\mu \vdash n}K_{\lambda \mu }K_{{\overline {\lambda }}\mu }$	1, 3, 9, 37, 177, 1054, 7237, 57447... (A321734) $a(n)=\sum _{\lambda ,\mu \vdash n}K_{\lambda \mu }^{2}$
Symmetric, nonincreasing row and column sums	1, 2, 5, 14, 39, 123, 393...	1, 3, 7, 21, 57, 182, 565, 1931, 6670... (A178718) $a(n)=\sum _{\lambda ,\mu \vdash n}K_{\lambda \mu }$

Others

Created

A347521: Number of polyominoes with n cells formed by coordinates that are not coprime.

A366020: Number of polyominoes in which every cell has coprime coordinates.

A327662: Length of shortest word of frequency depth n.

A340002: Random walk in R^3: Numerators of the expected distance after n steps.

A340003: Random walk in R^3: Denominators of the expected distance after n steps.

A360154: Primes of the form m^2 + 2*k^2 such that m^2 + 2*(k+1)^2 is also prime.

A360155: Primes of the form m^2 + 2*(k+1)^2 such that m^2 + 2*k^2 is also prime.

A360985: Number of full binary trees with n leaves, each internal node having the heights of its two subtrees weakly increasing left to right, and with k internal nodes having two subtrees of equal height.

A346797: Number of partitions of n into parts congruent to 0, 2 or 5 (mod 7).

A346798: Number of partitions of n into parts congruent to 0, 3 or 4 (mod 7).

A363685: number of descending plane partitions of order n with the sum of the parts equal to k.

Computed more terms

A308399: Number of partitions of n into parts congruent to 0, 3, or 5 (mod 8).

A082733: Sum of all entries in character table of the symmetric group S_n.

A094907: Number of different nontrivial two-digit cancellations of the form (xy)/(zx) = y/z in base n.

A317553: Sum of coefficients in the expansion of Sum_{y a composition of n} p(y) in terms of Schur functions, where p is power-sum symmetric functions.